084017 (2008) binary-black-hole initial data with nearly

Binary-black-hole initial data with nearly extremal spins

Geoffrey Lovelace,1,2 Robert Owen,1,2 Harald P. Pfeiffer,2 and Tony Chu2

1Center for Radiophysics and Space Research, Cornell University, Ithaca, New York 14853, USA2Theoretical Astrophysics 130-33, California Institute of Technology, Pasadena, California 91125, USA

(Received 27 May 2008; published 10 October 2008)

There is a significant possibility that astrophysical black holes with nearly extremal spins exist.

Numerical simulations of such systems require suitable initial data. In this paper, we examine three

methods of constructing binary-black-hole initial data, focusing on their ability to generate black holes

with nearly extremal spins: (i) Bowen-York initial data, including standard puncture data (based on

conformal flatness and Bowen-York extrinsic curvature), (ii) standard quasiequilibrium initial data (based

on the extended-conformal-thin-sandwich equations, conformal flatness, and maximal slicing), and

(iii) quasiequilibrium data based on the superposition of Kerr-Schild metrics. We find that the two

conformally flat methods (i) and (ii) perform similarly, with spins up to about 0.99 obtainable at the initial

time. However, in an evolution, we expect the spin to quickly relax to a significantly smaller value around

0.93 as the initial geometry relaxes. For quasiequilibrium superposed Kerr-Schild data [method (iii)], we

construct initial data with initial spins as large as 0.9997. We evolve superposed Kerr-Schild data sets with

spins of 0.93 and 0.97 and find that the spin drops by only a few parts in 104 during the initial relaxation;

therefore, we expect that superposed Kerr-Schild initial data will allow evolutions of binary black holes

with relaxed spins above 0.99. Along the way to these conclusions, we also present several secondary

results: the power-law coefficients with which the spin of puncture initial data approaches its maximal

possible value; approximate analytic solutions for large spin puncture data; embedding diagrams for single

spinning black holes in methods (i) and (ii); nonunique solutions for method (ii). All of the initial-data sets

that we construct contain subextremal black holes, and when we are able to push the spin of the excision

boundary surface into the superextremal regime, the excision surface is always enclosed by a second,

subextremal apparent horizon. The quasilocal spin is measured by using approximate rotational Killing

vectors, and the spin is also inferred from the extrema of the intrinsic scalar curvature of the apparent

horizon. Both approaches are found to give consistent results, with the approximate-Killing-vector spin

showing the least variation during the initial relaxation.

DOI: 10.1103/PhysRevD.78.084017 PACS numbers: 04.25.D, 02.70.Hm, 04.20.Ex, 04.25.dg

I. INTRODUCTION

There is a significant possibility that black holes withnearly extremal spins exist; by ‘‘nearly extremal,’’ wemean that the spin S and massM of the hole satisfy 0:95 &S=M2 & 1. Some models of black-hole accretion [1–3]predict that most black holes will have nearly extremalspins, and observational evidence for black holes withnearly extremal spins includes, e.g., estimates of black-hole spins in quasars [4] and estimates of the spin of a blackhole in a certain binary x-ray source [5]. There is consid-erable uncertainty about whether black holes do in facttypically have nearly extremal spins; e.g., some models [6–8] of black-hole accretion do not lead to large spins. Thisuncertainty could be reduced by measuring the holes’ spinsdirectly using gravitational waves.

This prospect of detecting the gravitational waves emit-ted by colliding black holes, possibly with nearly extremalspins, motivates the goal of simulating these spacetimesnumerically. Indeed, one focus of intense research has beenspinning black-hole binaries, including the discovery ofdramatic kicks when two spinning black holes merge [9–17] as well as some initial exploration of the orbital dy-

namics of spinning binaries [18–23]. All of these simula-tions start from puncture initial data as introduced byBrandt and Brugmann [24].The simplifying assumptions employed in puncture ini-

tial data make it impossible to construct black holes withspins arbitrarily close to unity. The numerical value of thefastest obtainable spin depends on which dimensionlessratio is chosen to characterize ‘‘black-hole spin.’’ Often,dimensionless spin is defined based on quasilocal proper-ties of the black hole

:¼ S

M2; (1)

where S is taken to be nonnegative and is a suitablequasilocal spin (e.g., obtained using approximate rotationalKilling vectors on the apparent horizon as described, forexample, in Appendix A) and M is a suitable quasilocalmass. The latter may be obtained from Christodoulou’sformula relating spin, area and mass of a Kerr black hole

M2 :¼ M2irr þ

S2

4M2irr

; (2)

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where we define the irreducible mass in terms of the area A

of the apparent horizon by Mirr :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA=16

p.

The quantity is not preserved during an evolution.Specifically, most black-hole initial data are not exactly inequilibrium, which leads to transients and emission of anartificial pulse of gravitational radiation early in numericalsimulations. The geometry in the vicinity of the black holesrelaxes on a time scale trelax (typically a fewM), and duringthis relaxation, the spin changes by

:¼ ðt ¼ 0Þ ðtrelaxÞ: (3)

When constructing a single spinning black hole with stan-dard puncture data [24], for instance, ðt ¼ 0Þ & 0:98,which seems encouragingly large. However Dain et al.[25,26] evolved standard puncture data with initial spinclose to this limit, and they find that the spin rapidly dropsto ðtrelaxÞ 0:93, i.e., 0:05.

For single-black-hole spacetimes, another widely useddimensionless spin measure is the ratio of total angularmomentum1 JADM and Arnowitt-Deser-Misner (ADM) en-ergy EADM:

"J :¼ JADME2ADM

: (4)

Dain et al. noted that ðtrelaxÞ is close to "J and explainedthis result as follows: The spacetime is axisymmetric,which implies that the angular momentum JADM is con-served and that the black hole’s spin equals JADM.Moreover, so long as a negligible fraction of the space-time’s energy is carried off by the spurious radiation, thehole’s quasilocal mass will relax to a value of EADM, givingðtrelaxÞ "J. Thus conformally flat Bowen-York datacannot be used to simulate black holes with nearly ex-tremal equilibrium spins, even though the initial spins canbe made fairly close to ¼ 1.

This paper examines three different approaches of con-structing black-hole initial data with nearly extremal spin.First, we revisit puncture initial data and inversion-symmetric Bowen-York initial data. We show that for asingle, spinning black hole at rest, both approaches areidentical, and we determine spin limits based purely oninitial data more accurately than before:

"J 0:928 200; ðt ¼ 0Þ 0:9837: (5)

We show that the limiting values of "J and ðt ¼ 0Þ areapproached as power laws of the spin parameter (curiously,with different powers). We furthermore give insight intothe geometric structure of these high-spin Bowen-Yorkinitial-data sets through numerical study and approximate

analytical solutions and find that a cylindrical throat formswhich lengthens logarithmically with the spin parameter.Second, we investigate the high-spin limit of another

popular approach of constructing initial data, the quasi-equilibrium formalism [27–31] based on the conformalthin sandwich equations [32,33]. For the standard choicesof conformal flatness and maximal slicing, we are able toconstruct initial data with spins somewhat larger than thestandard Bowen-York limits given in Eq. (5):

"J & 0:94; ðt ¼ 0Þ & 0:99: (6)

Once again "J is much lower than ðt ¼ 0Þ, which sug-gests that these data sets lead to equilibrium spins ofapproximate magnitude 0:94. Interestingly, thesefamilies of initial data are found to exhibit nonuniquesolutions [34–36], and the largest spins are obtained alongthe upper branch.The third approach also utilizes the quasiequilibrium

formalism [27–31], but this time we make use of the free-dom to choose arbitrary background data. Specifically, wechoose background data as a superposition of two Kerr-Schild metrics. This approach is based on the originalproposal of Matzner and collaborators [37,38] and wasfirst carried over into the conformal thin sandwich equa-tions in Ref. [39]; also, background data consisting of asingle, nonspinning Kerr-Schild black hole was used toconstruct initial data for a black-hole–neutron-star binaryin Ref. [40]. For single black holes, these data simplyreduce to the analytical Kerr solution. For binary blackholes, we construct initial data with spins as large as

ðt ¼ 0Þ ¼ 0:9997: (7)

We also present evolutions, demonstrating that our rapidlyspinning initial-data sets remain rapidly spinning after thenumerical evolution relaxes. In particular, we evolve anorbiting binary with ðt ¼ 0Þ ¼ 0:9275 and a head-onmerger with ðt ¼ 0Þ ¼ 0:9701. In both cases, j=ðt ¼0Þj is significantly smaller than 103. We conclude that theconformally curved superposed Kerr-Schild (SKS) initialdata we present in this paper, in contrast with conformallyflat Bowen-York data, are suitable for simulating binaryblack holes with nearly extremal spins.Throughout the paper, we use two different techniques

to measure the dimensionless spin of black holes, whichare described in the appendixes. The first (Appendix A)technique uses the standard surface-integral based on anapproximate rotational Killing vector of the apparent hori-zon. We compute the approximate Killing vector with avariation of the technique introduced by Cook and Whiting[41], extended with new normalization conditions of theapproximate Killing vector, and we denote the resultingspin ‘‘AKV spin’’ AKV. The second approach(Appendix B) is based on the shape of the horizon in theform of its scalar curvature; specifically, the spin magni-tudes are inferred from the minimum and maximum of the

1We define here JADM by an ADM-like surface integral atinfinity; in axisymmetry this definition coincides with the stan-dard Komar integral for angular momentum (see Sec. II B fordetails.)

LOVELACE, OWEN, PFEIFFER, AND CHU PHYSICAL REVIEW D 78, 084017 (2008)

084017-2

intrinsic Ricci scalar curvature of the horizon. We call thespin inferred in this way the ‘‘scalar-curvature spin,’’ andwe label the spin magnitudes inferred from the scalar-curvature minimum and maximum as min

SC and maxSC , re-

spectively. Typically, binary-black-hole initial data pro-duce holes that are initially not in equilibrium. Therefore,we use only the AKV spin to measure the initial black-holespin (Secs. III and IV.)We use both the AKVand the scalar-curvature spin when we measure the spin after the holeshave relaxed to equilibrium (Sec. V).

We also monitor whether any of the constructed initial-data sets have superextremal spins, as this may shed light,for example, on the cosmic censorship conjecture. Whenusing the Christodoulou formula [Eq. (2)] to define M, thequasilocal dimensionless spin is by definition bounded[42], 1. This can be seen most easily by introducingthe parameter , defined as

:¼ S

2M2irr

; (8)

and then rewriting as

¼ 1 ð1 Þ21þ 2

: (9)

The ratio is therefore not useful to diagnose super-extremal black holes. A more suitable diagnostic is foundin the parameter . For Kerr black holes, the first term onthe right-hand side of Eq. (2) is always smaller or equal tothe second, with equality only for extremal spin; i.e., 1, with equality for extremal spin. This motivates an alter-native definition of extremality [42]: A black hole is said tobe superextremal if the second term in Eq. (2) is larger thanthe first one, i.e., if > 1. In this paper, we monitor ,which we call the spin-extremality parameter, along withthe dimensionless spin . We find instances where ex-ceeds unity. Before this happens, however, a larger, sub-extremal ( < 1) apparent horizon appears, enclosing thesmaller, superextremal horizon (Sec. IVB, Fig. 12).

This paper is organized as follows. Section II summa-rizes the various formalisms that we use to construct initialdata. Section III investigates single-black-hole initial data,followed by the construction of binary-black-hole initialdata in Sec. IV. Section V presents binary-black-hole evo-lutions that show the good properties of superposed Kerr-Schild data and the various spin diagnostics. We summa-rize and discuss our results in Sec. VI. Finally,Appendixes A and B present our techniques to defineblack-hole spin.

II. INITIAL-DATA FORMALISM

Before constructing initial data for rapidly spinningsingle (Sec. III) and binary (Sec. IV) black holes, we firstsummarize the initial-data formalisms we will use. Afterlaying some general groundwork in Sec. II A, we describeBowen-York initial data (including puncture initial data) in

Sec. II B and quasiequilibrium extended-conformal-thin-sandwich data in Sec. II C.

A. Extrinsic curvature decomposition

Initial-data sets for Einstein’s equations are given on aspatial hypersurface and must satisfy the constraintequations

Rþ K2 KijKij ¼ 0; (10)

rjðKij gijKÞ ¼ 0: (11)

Here, gij is the induced metric of the slice , with cova-

riant derivative ri, R :¼ gijRij denotes the trace of the

Ricci tensor Rij, and Kij denotes the extrinsic curvature of

the slice as embedded into the spacetime manifold M.The constraint equations (10) and (11) can be trans-

formed into elliptic partial differential equations using aconformal transformation, e.g., [33]. One introduces aconformal metric ~gij via

gij ¼ c 4~gij; (12)

with the strictly positive conformal factor c > 0.Substituting Eq. (12) into Eq. (10) yields an elliptic equa-tion for c . One furthermore decomposes the extrinsiccurvature into trace and trace-free parts

Kij ¼ Aij þ 13g

ijK (13)

and splits off a longitudinal part from the trace-free extrin-sic curvature

Aij ¼ 1

ðLVÞij þMij: (14)

In Eq. (14), is a strictly positive weight function, the

longitudinal operator is defined as ðLVÞij ¼ 2rðiVjÞ 23g

ijrkVk, and Mij is symmetric and trace-free.2 Finally,

one introduces the conformally scaled quantities ¼ c 6 ~and Mij ¼ c10 ~Mij, which allows the momentum con-straint [Eq. (11)] to be rewritten completely in terms ofconformal quantities:

Aij ¼ c10 ~Aij; (15)

~A ij ¼ 1

~ð~LVÞij þ ~Mij: (16)

The Hamiltonian and momentum constraints then become

~r 2c 18~R 1

12K2c 5 þ 1

8~Aij

~Aijc7 ¼ 0; (17)

~rj

1

~ð~LVÞij

2

3c 6 ~riK þ ~rj

~Mij ¼ 0: (18)

2It is also possible, but not necessary, to require that Mij isdivergence-free.

BINARY-BLACK-HOLE INITIAL DATAWITH NEARLY- . . . PHYSICAL REVIEW D 78, 084017 (2008)

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Given choices for ~Mij, K, ~gij and ~, and also boundary

conditions, one can solve Eqs. (17) and (18) for c and Vi

and then assemble the (constraint-satisfying) initial datagij and Kij.

Many important approaches to construct binary-black-hole initial data can be cast in this form. The variousapproaches differ in the choices for the freely specifiableparts and the boundary conditions. Some choices of freedata aim for simplicity, such as Bowen-York initial data.Other approaches aim to preserve freedom, resulting inmore complicated sets of equations but also more flexibil-ity to control properties of the resulting initial data. Thequasiequilibrium extended-conformal-thin-sandwich ap-proach falls into this second category, and we will exploitprecisely its inherent freedom in choosing the free data toconstruct black holes with nearly extremal spins.

B. Bowen-York initial data

In this section, we describe two approaches of construct-ing initial data based on the well-known Bowen-Yorkextrinsic curvature. These two approaches, puncture dataand inversion-symmetric data, differ in how they treat thecoordinate singularity at r ¼ 0; both can be obtained fromthe general procedure outlined in Sec. II A by setting ~ 1, K 0, ~Mij 0 and by using a conformally flat metric

~g ij ¼ fij: (19)

The momentum constraint [Eq. (18)] then reduces to~rjð~LVÞij ¼ 0, which is solved by choosing the analytical

Bowen-York solutions [43,44].The Bowen-York solutions can be written down most

conveniently in Cartesian coordinates, fij ¼ ij:

ViP ¼ 1

4r½7Pi þ niPknk; (20)

ViS ¼ 1

r2ilmS

lnm; (21)

where r ¼ ðxixjijÞ1=2 is the coordinate distance to the

origin and ni ¼ xi=r is the coordinate unit vector pointingfrom the origin to the point under consideration. Thespatially constant vectors Pi and Si parametrize the solu-tions3

~AijP ¼ 3

2r2½2PðinjÞ ðij ninjÞPkn

k; (22)

~AijS ¼ 6

r3nðijÞklSknl: (23)

The conformal factor c is then determined by theHamiltonian constraint [Eq. (17)], which simplifies to

~r 2c þ 18c

7 ~Aij ~Aij ¼ 0: (24)

We would like to recover an asymptotically flat space; thisimplies the boundary condition c ! 1 as r ! 1.This boundary condition makes it possible to evaluate

the linear ADM-momentum and ADM-like angular mo-mentum of Bowen-York initial data without solvingEq. (24). These quantities are defined by surface integralsat infinity

JðÞ ¼ 1

8

I1ðKij gijKÞisjdA; (25)

where si is the outward-pointing unit normal to the inte-gration sphere.4 By letting c ! 1 in Eq. (15), one can

replace Kij by ~Aij and then evaluate the resulting integrals.

The choice of vector i determines which quantity iscomputed: For instance, ¼ ex corresponds to the x com-ponent of the linear ADM momentum; ¼ @ ¼ xey þyex yields the z component of the ADM-like angularmomentum.5 For Eqs. (22) and (23), the results arePiADM ¼ Pi and JiADM ¼ Si, respectively.The ADM energy is given by the expression

EADM ¼ 1

16

I1rjðGi

j ijGÞsidA; (26)

where Gij :¼ gij fij, G :¼ Gijgij. For conformal flat-

ness, Eq. (26) reduces to

EADM ¼ 1

2

I1@rc dA: (27)

The derivative of the conformal factor is known only afterEq. (24) is solved; therefore, in contrast with the linear andangular momenta, EADM can be computed only after solv-ing the Hamiltonian constraint.We now turn our attention to inner boundary conditions.

~AijP and ~Aij

S are singular at r ¼ 0. This singularity is inter-

preted as a second asymptotically flat universe; whensolving Eq. (24), this can be incorporated in two ways:

3In Cartesian coordinates, upper and lower indices are equiva-lent, so index positioning in Eqs. (20)–(23) is unimportant. Tofind ~Aij

P=S in another coordinate system, first compute theCartesian components Eqs. (20)–(23) and then apply the desiredcoordinate transformation.

4At infinity, the normal to the sphere si is identical to thecoordinate radial unit vector ni.

5As is common in the numerical relativity community, weintroduce the phrase ‘‘ADM angular momentum’’ to refer to anangular momentum defined at spatial infinity in the manner ofthe other conserved ADM quantities of asymptotically flat space-times [45], despite the fact that (at least to our knowledge), nosuch quantity is widely agreed to rigorously exist in general, dueto the supertranslation ambiguity that exists in four spacetimedimensions. For recent research on this issue see [46] andreferences therein. In the present paper, this subtlety can beignored, because we only compute this quantity in truly axisym-metric spacetimes, with ~ the global axisymmetry generator, sothat JADM coincides with the standard Komar integral for angularmomentum.


084017-4

(i) Inversion symmetry.—The demand that the solutionbe symmetric under inversion at a sphere with radiusRinv centered on the origin [44] results in a boundarycondition for c at r ¼ Rinv, namely, @c =@r ¼c =ð2RinvÞ. The Hamiltonian constraint Eq. (24)is solved only in the exterior of the sphere r Rinv, and the solution in the interior can be recoveredfrom inversion symmetry [44], e.g.,

c ðxiÞ ¼ Rinv

rc

R2inv

r2xi: (28)

(ii) Puncture data.—One demands [24] the appropriatesingular behavior of c for r ! 0 to ensure that thesecond asymptotically flat end is indeed flat. That is,c must behave as

c ðxiÞ ¼ mp

2rþ 1þ uðxiÞ (29)

for some positive parameter mp (the ‘‘puncture

mass’’) and function uðxiÞ that is finite and continu-ous inR3 and approaches 0 as r ! 1. Equation (24)then implies an equation for u that is finite every-where and can be solved without any inner bounda-ries:

~r 2u ¼ 1

8

~Aij~Aijr7

ðrþ mp

2 þ urÞ7 : (30)

The majority of binary black hole simulations usepuncture data; see, e.g., Refs. [9–23].

Both approaches allow specification of multiple blackholes at different locations, each with different spin andmomentum parameters Si and Pi. For puncture data this isalmost trivial; this accounts for the popularity of puncturedata as initial data for black-hole simulations. In contrast,for inversion-symmetric data, one needs to employ a rathercumbersome imaging procedure6 (see, e.g., [47] fordetails).

For a single spinning black hole at the origin, the ex-

trinsic curvature ~AijS given by Eq. (23) is identical for

inversion-symmetric and puncture data. For inversion-symmetric data, the conformal factor has the usual falloffat large radii:

c ðxiÞ ¼ 1þ EADM

2rþOðr2Þ; as r ! 1: (31)

Using Eq. (28) we find the behavior of c as r ! 0:

c ðxiÞ ¼ Rinv

rþ EADM

2Rinv

þOðrÞ; as r ! 0: (32)

Comparison with Eq. (29) shows that this is precisely the

desired behavior for puncture data, if one identifies Rinv ¼mp=2 and E=ð2RinvÞ ¼ 1þ uð0Þ. Because puncture data

have a unique solution, it follows that for single spinningblack holes, puncture data and inversion-symmetric dataare identical, provided mp ¼ 2Rinv.

For inversion-symmetric initial data for a single, spin-ning black hole, it is well known [48] that the apparenthorizon coincides with the inversion sphere: rAH ¼ Rinv.Therefore, we conclude that for puncture data for a single,spinning black hole, the apparent horizon is an exact

coordinate sphere with radius rAH ¼ mp=2, despite ~AijS

and uðxiÞ not being spherically symmetric.

C. Quasiequilibrium extended-conformal-thin-sandwich initial data

Another popular approach of constructing binary-black-hole initial data is the quasiequilibrium extended-confor-mal-thin-sandwich (QE-XCTS) formalism [27–31].Instead of emphasizing the extrinsic curvature, theconformal-thin-sandwich formalism [32] emphasizes thespatial metric gij and its time derivative. Nevertheless, it is

equivalent [33] to the extrinsic curvature decompositionoutlined in Sec. II A. The vector Vi is identified with theshift i,

Vi i; (33)

and the weight functions and ~ are identified (up tofactor 2) with the lapse and the conformal lapse, respec-tively,

2; ~ 2~: (34)

The tensor ~Mij is related to the time derivative of the

spatial metric ~uij :¼ @t~gij by

~M ij 1

2~~uij: (35)

Because Mij is trace-free [Eqs. (13), (15), and (16)], we

require ~uij to be trace-free.

The conformal-thin-sandwich equations allow control ofcertain time derivatives in the subsequent evolution of theconstructed initial data. If the lapse and shift i from theinitial data are used in the evolution, for instance, then thetrace-free part of @tgij will be proportional to ~uij.

Therefore (see Refs. [27,30])

~u ij 0 (36a)

is a preferred choice for initial-data sets that begin nearly inequilibrium, such as binary-black-hole quasicircular orbits.The evolution equation for K can be used to derive an

elliptic equation for the conformal lapse ~ (or, equiva-lently, for c ). Upon specification of

@tK 0; (36b)

6Even for a single black hole with Pk 0, Eq. (22) has to beaugmented by additional terms of Oðr4Þ to preserve inversionsymmetry [44].


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this fifth elliptic equation is to be solved for ~ simulta-neously with Eqs. (17) and (18); cf. [27,30].

Our numerical code uses the conformal factor c , theshift i, and the product of lapse and conformal factorc ¼ ~c 7 as independent variables, in order to simplifythe equation for @tK. Thus, the actual equations beingsolved take the form

0 ¼ ~r2c 18~Rc 1

12K2c 5 þ 1

8c7 ~Aij ~Aij; (37a)

0 ¼ ~rj

c 7

2ðc Þ ð~LÞij

2

3c 6 ~riK ~rj

c 7

2ðc Þ ~uij

;

(37b)

0 ¼ ~r2ðc Þ ðc Þ ~R8þ 5

12K4c 4 þ 7

8c8 ~Aij ~Aij

þ c 5ð@tK k@kKÞ; (37c)

with

~A ij ¼ c 7

2cðð~LÞij ~uijÞ: (37d)

These equations can be solved only after(1) specifying the remaining free data: i.e., the confor-

mal metric ~gij and the trace of the extrinsic curva-

ture K (we chose already ~uij 0 and @tK 0),

(2) choosing an inner boundary S which excises theblack holes’ singularities, and also an outer bound-ary B, and

(3) choosing boundary conditions for c , c , and i onB and S.

The initial data are required to be asymptotically flat,and the outer boundary B is placed at infinity.7 If ~gij is

asymptotically flat, the outer boundary conditions are then

c ¼ 1 on B; (38a)

c ¼ 1 on B; (38b)

i ¼ ð0 rÞi þ _a0ri on B: (38c)

Here ri is the coordinate position vector. The shift bound-ary condition consists of a rotation (parametrized by theorbital angular velocity 0) and an expansion (parame-trized by _a0); the initial radial velocity is necessary forreducing orbital eccentricity in binary-black-hole initialdata [49].

The inner boundary condition on the conformal factor censures that the excision surfaces S are apparent horizons[27]:

~sk@kc ¼ c3

8~~si~sj½ð~LÞij ~uij c

4~hij ~ri~sj

þ 1

6Kc 3 on S: (39)

Here ~si :¼ c 2si, si is the unit vector normal to S and~hij :¼ ~gij ~si~sj is the induced conformal 2-metric on S.The inner boundary condition on the shift is

i ¼ si ri on S; (40)

where isi ¼ 0. The first term on the right-hand sideensures that the apparent horizons are initially at rest; thetangential term determines the black hole’s spin [27–29].References [27–29] chose the sign of the last term in

Eq. (40) such that positive values ofr counteract the spinof the corotating holes that are obtained with r ¼ 0.Here, we are interested in large spins, and we reverse thesign of the last term in Eq. (40) so that positive, increasingr results in increasing spins.Two sets of choices for ~gij, K, S, and the boundary

condition for c on S are discussed in the next subsec-tions. Each set of choices will be used to construct binary-black-hole initial data in Sec. IV.

1. Conformal flatness and maximal slicing (CFMS)

The simplest choice for ~gij is a flat metric:

~g ij fij: (41)

This choice has been used almost exclusively in the pre-vious formulations of binary-black-hole initial data.The simplest choice for K, also commonly used in prior

formulations of binary-black-hole initial data, is maximalslicing, i.e.,

K 0: (42)

Also for simplicity, we choose to make the excisionsurface S consist of coordinate spheres:

S ¼ [na¼1

Sa; (43)

where Sa are surfaces of constant Euclidean distance rexcabout the center of each excised hole, and n ¼ 1 or 2 is thenumber of black holes present in the initial data.The boundary condition for the lapse on S determines

the temporal gauge; we adopt the condition given inEq. (59a) of Ref. [28]:

@

@raðc Þ ¼ 0 on Sa; (44)

where ra is the Euclidean distance from the center of holea. This type of initial data is used in Refs. [49,65,66].

2. Superposed Kerr-Schild

Single black holes with angular [50,51] or linear [52]momentum do not admit conformally flat spatial slicings;therefore, conformal flatness [Eq. (41)] is necessarily de-ficient. This has motivated investigations of binary-black-hole initial data whose free data have stronger physicalmotivation, e.g., Refs. [37,38,53–59].

7In practice, B is a sphere with radius * 109 times thecoordinate radius of the black-hole horizons.


084017-6

In this subsection, we consider conformally curved datathat are in the same spirit as the SKS data of Refs. [37,38]although here (i) we apply the idea to the QE-XCTSformalism, and (ii) as discussed below, our free data isvery nearly conformally flat and maximally sliced every-where except in the vicinity of the black holes.

The choices we make here generalize the conformallycurved data in Chapter 6 of Ref. [39] to nonzero spins.Specifically, the free data and lapse boundary conditionwill be chosen so that the conformal geometry near eachhole’s horizon is that of a boosted, spinning, Kerr-Schildblack hole. The conformal metric ~gij and the mean curva-

ture K take the form

~g ij :¼ fij þXna¼1

er2a=w2aðgaij fijÞ; (45)

K :¼ Xna¼1

er2a=w2aKa: (46)

Here gaij and Ka are the spatial metric and mean curvature,

respectively, of a boosted, spinning Kerr-Schild black hole

with mass ~Ma, spin ~Sa, and speed ~va.Far from each hole’s horizon, the conformal metric is

very nearly flat; this prevents the conformal factor fromdiverging on the outer boundary [39]. The parameter wa isa weighting factor that determines how quickly the curvedparts of the conformal data decay with Euclidean distancera (a ¼ 1; 2; . . . ) from hole a; in this paper, the weightfactor wa is chosen to be larger than the size scale of hole abut smaller than the distance d to the companion hole (ifany): Ma & wa & da. This is similar to the ‘‘attenuated’’

superposed Kerr-Schild data of Refs. [38,60], except thathere the weighting functions are Gaussians which vanishfar from the holes, while in Refs. [38,60] the weightingfunctions go to unity far from the holes.The excision surfaces Sa are not coordinate spheres

unless ~Sa ¼ 0 and ~va ¼ 0. Instead they are deformed intwo ways. (i) They are distorted so that they are surfaces ofconstant Kerr radius rKerr, i.e.,

x2 þ y2

r2Kerr þ ~Sa2= ~Ma

2þ z2

r2Kerr¼ 1; (47)

where x, y, and z are Cartesian coordinates on the S. Then,(ii) the excision surfaces are Lorentz-contracted along thedirection of the boost.The boundary condition for the lapse on Sa is a

Dirichlet condition that causes (and, consequently, thetemporal gauge) in the vicinity of each hole to be nearlythat of the corresponding Kerr-Schild spacetime, i.e.,

c ¼ 1þ Xna¼1

er2a=w2aða 1Þ on Sa; (48)

where a is the lapse corresponding to the Kerr-Schildspacetime a.

III. SINGLE-BLACK-HOLE INITIAL DATAWITHNEARLY EXTREMAL SPINS

In this section, we examine to which extent the formal-isms presented in Sec. II can generate single-black-holeinitial data with nearly extremal spin. We consider firstBowen-York initial data and then conformally flat quasi-

TABLE I. Summary of the initial-data sets constructed in this paper. The first row (BY-Single) represents Bowen-York initial data forsingle black holes of various spins. The next two rows (CFMS-Single and CFMS) are quasiequilibrium, conformally flat, maximallysliced initial data for single and binary spinning black holes, respectively. All other data sets employ superposed Kerr-Schildquasiequilibrium data with the second block of rows representing families of initial-data sets for various spins and the last block ofrows representing individual data sets to be evolved. The data sets SKS-0.93-E0 to SKS-0.93-E3 demonstrate eccentricity removal, andSKS-Headon is used in a head-on evolution. The first block of columns gives the label used for each data set and the relevant section ofthis paper devoted to it. The next block of columns lists the most important parameters entering the initial data. The last block ofcolumns lists some properties of those data sets that we evolve in Sec. V.

Label Section Figures n d 0 _a0 104 r or S=m2p

~S jAKVj Mirr M EADM

BY-Single III A 1–5, 8, and 19 1 0:01 S=m2p 104

CFMS-Single III B 6–8 and 19 1 0 r 0:191 CFMS IVA 9 and 13 2 32 0.007 985 0 0 r 0:1615 SKS-0.0 IVB 11 and 13 2 32 0.006 787 0 0 r 0:24 0

SKS-0.5 IVB 11 and 13 2 32 0.006 787 0 0 r 0:27 0.5

SKS-0.93 IVB 11–13 2 32 0.006 787 0 0 r 0:35 0.93

SKS-0.99 IVB 10–13 2 32 0.007 002 3.332 0:28 r 0:39 0.99

SKS-0.93-E0 VB 14 2 32 0.006 787 0 0.28 0.93 0.9278 0.9371 1.131 2.243

SKS-0.93-E1 VB 14 2 32 0.007 0 0.28 0.93 0.9284 0.9375 1.132 2.247

SKS-0.93-E2 VB 14 2 32 0.006 977 3.084 0.28 0.93 0.9275 0.9395 1.134 2.249

SKS-0.93-E3 VC 10, 11, 13–16, and 19 2 32 0.007 002 3.332 0.28 0.93 0.9275 0.9397 1.134 2.250

SKS-Headon VD 10, 11, 13, and 17–19 2 100 0 0 0.3418 0.97 0.9701 0.8943 1.135 2.257


084017-7

equilibrium data. Since superposed Kerr-Schild data canrepresent single Kerr black holes exactly, there is no needto investigate single-hole superposed Kerr-Schild data. InSec. IV, we will both consider conformally flat and super-posed Kerr-Schild data for binary black holes.

To orient the reader, the initial-data sets constructed inthis section, as well as the binary-black-hole data setsconstructed in Sec. IV, are summarized in Table I.

Unless noted otherwise, all spins presented in this sec-tion are measured using the approximate-Killing-vectorspin AKV described in Appendix A. Therefore, the sub-script ‘‘AKV’’ in AKV will be suppressed for simplicity.

A. Bowen-York (puncture) initial data

As discussed in Sec. II B, for a single spinning blackhole at rest, puncture initial data are identical to inversion-symmetric initial data. Such solutions have been examinedin the past (e.g., [48,61]), and additional results wereobtained (partly in parallel to this work) in the study byDain, Lousto, and Zlochower [25].

We revisit this topic here to determine the maximumpossible spin of Bowen-York initial data more accuratelythan before, to establish the power-law coefficients for theapproach to these limits with increasing spin parameter S,and to present new results about the geometric structure ofBowen-York (BY) initial data with a very large spinparameter.

We solve Eq. (30) with the pseudospectral elliptic solverdescribed in Ref. [62]. The singular point of u at the originis covered by a small rectangular block extending from104mp along each coordinate axis. This block overlaps

four concentric spherical shells with radii of the boundariesat 8 105mp, 0:005mp, 0:3mp, 50mp, and 109mp. The

equations are solved at several different resolutions, withthe highest resolution using 203 basis functions in the cube,L ¼ 18 in the spheres and 26 and 19 radial basis functionsin the inner and outer two spherical shells, respectively.

Because of the axisymmetry of the data set, the rota-tional Killing vector of the apparent horizon is simply @.

The integral for the quasilocal spin Eq. (A1) turns out to beindependent of c and can be evaluated analytically with aresult equal to the spin parameter S. Thus we can use thisinitial-data set to check how well our spin diagnostics andour ADM angular momentum diagnostic works (recall thatJADM is also equal to the spin parameter S). This compari-son is performed in Fig. 1, which shows relative differencesbetween the numerically extracted values for the AKVspin, the coordinate spin (defined with the AKV spin inAppendix A), and the ADM angular momentum JADMrelative to the expected answer S. The figure also showsdifferences between neighboring resolutions for the twoquantities of interest below: S=M2 ¼ and S=E2

ADM ¼JADM=E

2ADM ¼ "J.

Figure 1 seems to show exponential convergence withincreased resolution N. Since puncture data are only C2 at

the puncture, one would rather expect polynomial conver-gence. The effect of the nonsmoothness at the puncture ismitigated by choosing a very high resolution close to thepuncture (a small cube with sides104mp with 203 basis

functions). Therefore, for the resolutions considered inFig. 1, the truncation error is dominated by the solutionaway from the puncture, and exponential convergence isvisible. If we used infinite-precision arithmetic and werepushing toward higher resolution than shown in Fig. 1, thenwewould expect to eventually see polynomial convergencedominated by the cube covering the puncture.Next, we construct a series of initial-data sets with

increasing spin parameter S and compute , "J, and for each initial-data set. The results are plotted in Fig. 2and confirm earlier results [26,61]. In addition, the insetshows that the asymptotic values max ¼ 0:9837 and"u;max ¼ 0:928200 are approached as power laws in the

spin parameter:

max /S

m2p

0:75; (49)

"J;max "J /S

m2p

1:4: (50)

16 24 32 40

10-10

10-8

10-6

10-4

10-2

100

N1/3

SCoord

S/EADM

2

SAKV

JADM

S/M2

Single puncture BH, S/mp

2=10000

FIG. 1 (color online). Convergence test for a single punctureblack hole with a very large spin parameter S=m2

p ¼ 10 000.

Plotted are results vs resolution N, which is the total number ofbasis functions. The solid lines show the relative differences ofthree angular momentum measures to the analytically expectedvalue 10 000. The dashed lines show differences from the next-higher resolution of two dimensionless quantities for which noanalytic answer is available.


084017-8

The exponents of these power laws are computed here forthe first time.

To confirm that the apparent horizon is indeed at r ¼Rinv, we ran our apparent horizon finder on the high-spinpuncture initial-data sets. The horizon finder had greatdifficulty converging, and the reason for this becomes clearfrom Fig. 3. The main panel of this figure shows the area ofspheres with coordinate radius r. The area is minimal atr ¼ mp=2, as it must be, sincemp=2 ¼ Rinv is the radius of

the inversion sphere. However, the area is almost constantover a wide range in r—for S=m2

p ¼ 10 000 over about two

decades in either direction: 0:01 & r=Rinv & 100. Thus,the Einstein-Rosen bridge (the throat) connecting the twoasymptotically flat universes lengthens as the spin in-creases, giving rise to an ever-lengthening cylinder. Ifthis were a perfect cylinder, then the expansion would bezero for any r ¼ const cross section. Because the geometryis not perfectly cylindrical, the expansion vanishes only forr ¼ mp=2 ¼ Rinv, but remains very small even a signifi-

cant distance away from r ¼ mp=2 ¼ Rinv. This is shown

in the inset, which plots the residual of the apparent hori-zon finder at different radii.

With the lengthening of the throat, the interval in r withsmall expansion lengthens, and the value of the expansionwithin this interval reduces. Both effects make it harder for

the apparent horizon finder to converge. In Fig. 2, we haveused our knowledge of the location of the apparent horizonto set rAH ¼ mp=2 rather than to find this surface numeri-

cally. Without this knowledge, which arises due to theidentification of puncture data and inversion-symmetricdata, computation of Fig. 2 would have been significantlyharder, perhaps impossible.Let us assume for the moment that the solution c ðrÞ ¼

mp

2r þ 1þ uðrÞ is spherically symmetric (we give numerical

evidence below that this is indeed a good approximation).Because gij ¼ c 4fij, the area of coordinate spheres is then

given by

AðrÞ ¼ 4c 2ðrÞr: (51)

In the throat region, where AðrÞ const, the conformalfactor must therefore behave like 1=

ffiffiffir

p, as also argued

independently by Dain, Lousto, and Zlochower [25].To extend on Dain, Lousto, and Zlochower’s analysis, let

us substitute Eq. (23) into Eq. (24) to obtain the well-known equation

~r 2c ¼ 9S2sin2

4r6c7; (52)

where is the angle between the spin direction and thepoint xi. Continuing to assume that c is approximately

0.01 0.1 1 10 100 1000 100000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 100 10000

10-6

10-4

10-2

χmax

- χε

J max - ε

J

χε

J

n=-1.4

n=-0.75

S/mp2

ζ

FIG. 2 (color online). Properties of single, spinning punctureblack holes with spin parameter S and puncture mass mp. The

dimensionless spin :¼ S=M2, ADM angular momentum "J :¼JADM=E

2ADM, and spin-extremality parameter :¼ S=ð2M2

irrÞ areplotted against the spin parameter S=m2

p. The horizon mass M is

related to the spin S and irreducible mass Mirr in Eq. (2).

0.001 0.01 0.1 1 10 100 100010

4

105

106

107

108

109

0.1 1 1010

-8

10-6

10-4

10-2

2r /mp = r /Rinv

Res AH

Area A(r)/Rinv

2

S/mp

2=10000

1000100

100001000100

FIG. 3 (color online). Properties of coordinate spheres withradius r for high-spin puncture initial data. Main panel: Area ofthese spheres. Inset: Residual of the apparent horizon equationon these spheres. The area is almost constant over several ordersof magnitude in r. The apparent-horizon residual vanishes at r ¼Rinv but is very small over a wide range of r.


084017-9

spherically symmetric, we can replace the factor sin2 byits angular average ð4Þ1

Rsin2d ¼ 2=3 and obtain

d2 c

dr2þ 2

r

d c

dr¼ 3S2

2r6c7: (53)

Here, we introduced an overbar c to distinguish the spheri-cally symmetric solution c ðrÞ of Eq. (53) from the fullsolution c ðxiÞ of puncture/inversion-symmetric initialdata. Following Dain, Lousto, and Zlochower [25] weassume that the conformal factor behaves as a power law[ c ðrÞ ¼ Ar] and substitute this into Eq. (53). We find thatEq. (53) determines the power-law exponent ¼ 1=2

and the overall amplitude A ¼ ð6S2Þ1=8, so that

c ðrÞ ¼ ð6S2Þ1=8ffiffiffir

p ¼ 961=8S

m2p

1=4

r

Rinv

1=2: (54)

In Eq. (54), we chose the scaling S=m2p, which is com-

monly used in the puncture-data literature, but kept r=Rinv

to emphasize the inversion symmetry of the data in ourfigures (in a log-plot using r=Rinv, the solution will appearsymmetric; see, e.g., Fig. 3). While c ðrÞ solves the spheri-cally symmetric Eq. (53) exactly, it must deviate fromc ðxiÞ for sufficiently large r because c ! 0 as r ! 1,whereas c ! 1. The deviation will become significant

when c 1, i.e., at radius rx ffiffiffiffiffiffiffiffiffiffiffiffiS=m2

p

q. Because of in-

version symmetry, this implies a lower bound of validity at1=rx, so that Eq. (54) holds for

S

m2p

1=2&

r

Rinv

&

S

m2p

1=2

: (55)

The circumference of the cylindrical throat is

C ¼ 2 c ðrÞ2r ¼ 2961=4ffiffiffiffiffiffiffiS

m2p

sRinv; (56)

and its length is

L ¼Z ðS=m2

pÞ1=2

ðS=m2pÞ1=2

c 2ðrÞdr ¼ 961=4ffiffiffiffiffiffiffiS

m2p

sln

S

m2p

Rinv: (57)

Therefore, the ratio of length to circumference

LC

¼ 1

2ln

S

m2p

(58)

grows without bound as S=m2p becomes large, albeit very

slowly. The scaling with ðS=m2pÞ1=2 in Eqs. (55)–(57) might

seem somewhat surprising. However, in the large spinlimit, S=M2 is just a constant close to unity (namely,

max ¼ 0:9837). Therefore, S1=2 M, i.e., the scaling

S1=2 is effectively merely a scaling with mass.Figure 4 shows the conformal factor c , the ‘‘puncture

function’’ u, and the estimate c of Eq. (54) for three

different values of S=m2p. There are several noteworthy

features in this figure. First, both c and u show clearlythree different regimes:(i) For large r, c 1 and u / 1=r. This is the upper

asymptotically flat end.(ii) For intermediate r, c / 1=

ffiffiffir

pand u / 1=

ffiffiffir

p. This

is the cylindrical geometry extending symmetricallyaround the throat. This region becomes more pro-nounced as S increases.

(iii) For small r, c / 1=r and u const. This is thelower asymptotically flat end.

Figure 4 also plots the approximate solution c [cf.Eq. (54)] for its range of validity [given by Eq. (55)].Note that slope and amplitude of c fit very well thenumerical solution c . In fact, the agreement is much betterthan with u.One could also have started the calculation that led to

Eq. (54) with Eq. (30). Assuming spherical symmetry, andassuming that u mp=ð2rÞ þ 1, we would have derived

Eq. (53), but with c replaced by u. We would then havefound the approximate behavior Eq. (54) for u. The dis-advantage of this approach is the need for additional ap-proximations, which reduce the accuracy of the result.From Fig. 4 we see that, in the throat region, the dotted

0.001 0.01 0.1 1 10 100 1000

0.1

1

10

100

1000ψuΨ

2r/mp = r /R

inv

S/mp

2=10000

1000100

FIG. 4 (color online). Solutions of high-spin puncture initialdata. Plotted are the conformal factor c and puncture function uin the equatorial plane as a function of radius r. Furthermore, theapproximate solution c is included, with solid circles denotingthe range of validity of this approximation; cf. Eq. (55). Threecurves each are plotted, corresponding from top to bottom toS=m2

p ¼ 10 000, 1000, and 100.


084017-10

lines representing c are close to the dashed lines of u. Butthe agreement between c and c is certainly better.

Finally, we note that the limits of validity of c [Eq. (55)]match very nicely the points where the numerical c di-verges from c .

To close this section, we present numerical evidence thatindeed c is approximately spherically symmetric, theassumption that entered into our derivation of Eq. (54).We decompose the conformal factor of the numericalpuncture-data solutions into spherical harmonics,

c ðr; ;Þ ¼ X1l¼0

Xlm¼l

c lmðrÞYlmð;Þ; (59)

and plot in Fig. 5 the sizes of the l 0 modes relative tothe spherically symmetric mode c 00. Because of the sym-metries of the problem, the only nonzero modes have m ¼0 and even l. In the throat region, the largest non-spherically symmetric mode c 20 is about a factor of 65smaller than the spherically symmetric mode. With in-creasing l, c lm decays very rapidly. Also, in both asymp-totically flat ends, the non-spherically symmetric modesdecay more rapidly than the l ¼ 0 mode, as expected forasymptotically flat data. This figure again shows nicely theinversion symmetry of the data, under r=Rinv !ðr=RinvÞ1. Given the simple structure of the higher modes,it should be possible to extend the analytical analysis of thethroat to include the nonspherical contributions. To do so,one would expand c as a series in Legendre polynomialsin ; the c7 term on the right-hand side of Eq. (52) wouldresult in a set of ordinary differential equations for those

coefficients. In the throat region, the radial behavior ofeach mode should be / 1=

ffiffiffir

p, and the ordinary differential

equations should simplify to algebraic relations.

B. Quasiequilibrium extended-conformal-thin-sandwich data

We have seen in Sec. III A that puncture initial data forsingle, spinning black holes can be constructed for holeswith initial spins of 0:9837. In this section, we addressthe analogous question for excision black-hole initial data:How rapid can the initial spin be for a single, spinningblack hole constructed using QE-XCTS initial data?As noted previously, if the free data ~gij andK are chosen

to agree with the analytic values for a Kerr black hole, gKerrij

and KKerr, then the QE-XCTS initial data can exactlyrepresent a single Kerr black hole. In this case, ¼ 1 is

obtained trivially by choosing ~S ¼ ~M2 ¼ 1, where ~M and~S are the mass and spin, respectively, of the Kerr black holedescribed by the conformal metric.Setting aside this trivial solution, we construct CFMS

data for a single, spinning hole. We construct a family ofQE-XCTS initial-data sets for single spinning black holesby numerically solving the XCTS equations [in the formstated in Eqs. (37a)–(37c)] using the same spectral ellipticsolver [62] as in Sec. III A. The free data are given byEqs. (41) and (42) and by Eqs. (36a) and (36b).On the outer boundary B, we impose Eqs. (38a)–(38c).

So that the coordinates are asymptotically inertial, wechoose 0 ¼ _a0 ¼ 0 in Eq. (38c).We excise a coordinate sphere of radius rexc about the

origin, where

rexc ¼ 0:859 499 77 (60)

is chosen such that for zero spin M ¼ 1. On this innerboundary S, we impose Eqs. (39), (40), and (44). The spinis determined by Eq. (40): First, the vector i is chosen tobe the coordinate rotation vector @, making the spin point

along the positive z axis; then, the rotation parameterr isvaried while the other parameters are held fixed. The spinis measured on the apparent horizon using theapproximate-Killing-vector spin (Appendix A); becausein this case the space is axisymmetric, the ‘‘approximate’’Killing vector reduces to the corresponding exact rota-tional Killing vector.Figure 6 show how the massM and AKV spin S depend

on r. At r ¼ 0, we find the spherically symmetricsolution with S ¼ 0 and Mirr ¼ M ¼ 1 [the mass is pro-portional to the excision radius, and Eq. (60) sets it tounity]. Using this spherically symmetric solution as aninitial guess for the elliptic solver, we find solutions forincreasing r with spin increasing initially linearly withr and with approximately constant mass. Beyond somecritical r;crit, the elliptic solver fails to converge, and

close to this point, all quantities vary in proportion to

0.001 0.01 0.1 1 10 100 100010

-7

10-6

10-5

10-4

10-3

10-2

10-1 S=10000

S=1000S=100

|Ψlm

| / Ψ00

2r/mp = r/R

inv

l=2, m=0

l=4, m=0

l=6, m=0

l=8, m=0

FIG. 5 (color online). Angular decomposition of the conformalfactor c ðr; ;Þ for single-black-hole puncture data.


084017-11

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir;crit r

p. These symptoms indicate a critical point

where the solutions ‘‘turn over’’ and continue towardssmaller r. Analogous nonunique solutions of the XCTSequations have been discovered before in Ref. [34]. Toconstruct solutions along the upper branch, one mustchoose a sufficiently close initial guess for the ellipticsolver; we follow the steps outlined in Ref. [34] and areable to find solutions along the upper branch for a widerange ofr <r;crit. As Fig. 6 shows, mass and spin of the

horizon in solutions along the upper branch increase withdecreasing r, analogous to the findings in [34,35].

Figure 7 shows the dependence of ¼ S=M2, "J ¼JADM=E

2ADM, and ¼ S=ð2M2

irrÞ on r. The curves reflect

again the nonunique solutions. The dimensionless spin increases continuously along the lower branch and reaches 0:85 at the critical point. Asr is decreased along theupper branch, continues to increase, eventually reachingvalues larger than 0.99. It appears continues to increaseas r ! 0. To find the limiting value, consider that thebehavior of the extremality parameter in the inset ofFig. 7. Assuming that can be extrapolated to r ! 0,we find a limiting value of 0:88. By Eq. (9), thisimplies a maximal value of 0:992.

In Figs. 6 and 7, the data sets on the lower branch appearto be physically reasonable. For spins & 0:85, the mass

M is nearly constant, and the dimensionless spin in-creases linearly with r. Furthermore, as r ! 0 thelower branch continuously approaches the exactSchwarzschild spacetime (see [28]). The upper branchappears to be physically less reasonable; for instance, thespin increases for decreasing horizon frequency r.Comparing Figs. 2 and 7, we see that the QE-XCTS datalead to somewhat larger values of and "J relative topuncture data. However, the values are not too different,and similar trends remain. For instance, is much closer tounity than "J.To investigate differences or similarities between punc-

ture data and QE-XCTS data further, we compute embed-ding diagrams of the equatorial planes of these data sets.The initial data for single black holes have rotationalsymmetry about the z axis, so the metric (12) on theinitial-data hypersurface, when restricted to the equatorialplane, can be written as

ds2 ¼ c 4ðdr2 þ r2d2Þ; (61)

where r and are the usual polar coordinates. This metricis now required to equal the induced metric on the 2Dsurface given by Z ¼ ZðRÞ embedded in a 3D Euclideanspace with line element

0 0.05 0.1 0.15 0.2Ω

r

0.1

1

10

100

1000

0.18 0.185 0.19

1.5

2

2.5

S

M

Mirr

FIG. 6 (color online). Conformally flat, maximally sliced,quasiequilibrium initial-data sets with a single, spinning blackhole. We plot the horizon massM, irreducible massMirr, and the(approximate-Killing-vector) spin S against the rotation parame-ter r [cf. Eq. (40)]. Only r is varied in this figure; all otherparameters are held fixed. The upper and lower points with thesame r are obtained numerically by choosing different initialguesses. The inset shows a close-up view of the turning point,which occurs at r 0:191.

0 0.05 0.1 0.15 0.2Ω

r

0

0.2

0.4

0.6

0.8

0 0.03 0.06

0.85

0.9

0.95

1

χε

J

ζ

FIG. 7 (color online). Conformally flat, maximally sliced qua-siequilibrium initial-data sets with a single spinning black hole:The dimensionless spin , dimensionless ADM angular momen-tum "J, and spin-extremality parameter plotted against r [cf.Eq. (40)]. Only r is varied in this figure; all other parametersare held fixed. The inset enlarges the area in the upper leftcorner; we are able to generate data sets with > 0:99, whereasthe largest spin obtainable on the lower branch is 0:85.


084017-12

ds2Euclidean ¼ dR2 þ R2d2 þ dZ2: (62)

Setting dZ ¼ dZdR dR, we obtain the induced metric on the

Z ¼ ZðRÞ surface

ds2 ¼1þ

dZ

dR

2dR2 þ R2d2: (63)

Equating Eqs. (61) and (63), we find

R ¼ c 2r (64)

and 1þ

dZ

dR

2dR2 ¼ c 4dr2: (65)

Combining (64) and (65) results indZ

dr

2 ¼ 4rc 2 dc

dr

c þ r

dc

dr

: (66)

Since the pseudospectral elliptic solver gives c as a func-tion of r, Eqs. (64) and (66) allow us to solve for theembedding radius R and the embedding height Z in termsof r.

Figure 8 shows embedding diagrams for three sets ofQE-XCTS and puncture data. We have set Z ¼ 0 at r ¼rexc for QE-XCTS data and at r ¼ Rinv for puncture data.This figure also contains the embedding of a plane throughSchwarzschild in Schwarzschild coordinates (i.e., the S ¼

0 limit of BY puncture data), given by R=M ¼Z2=ð8M2Þ þ 2. Both puncture data and CFMS data exhibita lengthening throat with increasing spin S=M2. For punc-ture data, this lengthening can be deduced from the ana-lytical results in Sec. III A: As the spin parameter S of thepuncture data increases by a factor of 10 while mp 1 is

held constant, we find from Eq. (57) that L=S1=2 shouldincrease by

L=S1=2 ¼ 961=4

2ln10 3:60; (67)

where the factor 1=2 arises because Rinv ¼ mp=2 ¼ 0:5.

The embedding diagram shows only the top half of the

throat, and S1=2 M [cf. the discussion after Eq. (58)].Therefore in Fig. 8 the S ¼ 100, 1000, and 10 000 lines forBY (puncture) data should be spaced by Z=M 1:80 forlarge R=M. This indeed is the case.The CFMS data sets appear to scale proportionally toffiffiffiS

p, which is similar to the puncture data’s behavior.

Furthermore, the CFMS initial-data sets also develop alengthening throat as S becomes large (the effect is notas pronounced as for puncture data, owing to the smallermaximal S we achieved). Thus it appears that large spinCFMS data might be similar to large spin puncture data.However, the throats of the QE-XCTS data show a bulgenear the bottom, because for these data sets R actuallydecreases with r in the immediate vicinity of rexc. This isunlike the puncture data, which very clearly exhibit cylin-drical throats, consistent with the discussion leading to(58).

0 4 8 12 16R/M

0

4

8

12

16

Z/M

BY, S=10000BY, S=1000BY, S=100Ω

r=0.016, S=1034

Ωr=0.05, S=100.8

Ωr=0.135, S=10.6

1.6 1.7 1.80

1

2

3

FIG. 8 (color online). Embedding diagrams for puncture andquasiequilibrium initial data. Plotted is the embedding height Zas a function of the embedding radius R, both scaled by the massM. For quasiequilibrium data (dashed lines), Z ¼ 0 at r ¼ rexc;for puncture data (solid lines), Z ¼ 0 at r ¼ Rinv. The thin solidpurple curve represents the embedding of a plane through aSchwarzschild black hole in Schwarzschild slicing.

0 0.04 0.08 0.12

0.96

0.98

1

0 0.05 0.1 0.15Ω

r

0

0.2

0.4

0.6

0.8

1

ζ

χ

FIG. 9 (color online). Main panel: Dimensionless spin [Eq. (1)] and spin-extremality parameter [Eq. (8)] for thefamily CFMS of spinning binary-black-hole initial data.Inset: Enlargement of toward the end of the upper branch,with circles denoting the individual initial-data sets that wereconstructed. Compare with Fig. 7.


084017-13

IV. BINARY-BLACK-HOLE INITIAL DATAWITHNEARLY EXTREMAL SPINS

In this section, we construct binary-black-hole initialdata with rapid spins, confining our attention to the specialcase of spins aligned with the orbital angular momentum.In the limit of large separation, binary-black-hole punctureinitial data will behave like two individual puncture initial-data sets. Specifically, we expect that it should be possibleto construct puncture binary-black-hole initial data withinitial spins ðt ¼ 0Þ & 0:98, but the spins will rapidlydrop to & 0:93 as the black holes settle down. For thisreason, and also because puncture data are not well-suitedto our pseudospectral evolution code, we will restrict ourattention to binary black holes constructed with the QE-XCTS approach.

As laid out in Table I, we first construct a family (labeledCFMS) of standard conformally flat initial data on maxi-mal slices; then, we turn our attention to families (labeledSKS) of superposed Kerr-Schild initial data. Finally, weconstruct a few individual SKS-initial-data sets which weevolve in Sec. V. All of the data sets represent equal-mass,equal-spin black holes with spins parallel to the orbitalangular momentum.

In this section, unless otherwise indicated, all dimen-sionless spins are the approximate-Killing-vector spinAKV (Appendix A), and the subscript AKV will be sup-pressed for simplicity.

A. Conformally flat, maximal slicing data

To construct conformally flat binary-black-hole data, wesolve the same equations and boundary conditions as forthe single-black-hole case, as described in Sec. III B, withthe main difference being that we excise two spheres withradius rexc [cf. Eq. (60)] with centers on the x axis at x ¼d=2. The initial spins of the holes are set by adjustingr,just as in the single-hole case. The parameters0 and _a0 inthe outer boundary condition on the shift [Eq. (38c)] de-termine the initial angular and radial motion of the holes,which in turn determine the initial eccentricity e of theorbit. We set 0 ¼ 0ez, where ez is a unit vector thatpoints along the positive z axis. For the CFMS family ofdata sets considered here, we use values for0 and _a0 thatshould result in closed, fairly circular orbits, since ourchoices of 0 and _a0 lead to data sets that approximatelysatisfy the Komar-mass condition EADM ¼ MK (cf. [29]).Specifically, on the lower branch of the resulting nonun-ique family of initial data,

jEADM MKjEADM

& 1%; (68)

where the Komar mass is defined by [e.g., Eq. (35) ofRef. [29]]

MK :¼ 1

4

I1ðri jKijÞdA: (69)

(On the upper branch, EADM and MK differ by up to 3%.)

As the rotation parameter r is varied (with the coor-dinate separation d held fixed), we find that the CFMSfamily of binary-black-hole initial data behaves qualita-tively similarly to the analogous single-black-hole initialdata discussed in Sec. III B. There is a maximalr;crit such

that no solutions can be found for r >r;crit; for values

of r below r;crit, two solutions exist. Figure 9 plots the

dimensionless spin and the spin-extremality parameter against r for this family of initial data. We only showvalues for one of the holes, since the masses and spins areequal. Spins larger than 0:85 appear on the upperbranch. The highest spin we have been able to constructis larger than ¼ 0:97.

B. Superposed Kerr-Schild data

In this section, we solve the same equations and bound-ary conditions as in the conformally flat case, except thatwe use SKS free data (Sec. II C 2) instead of conformallyflat free data. To construct the individual Kerr-Schild data,we need to choose for each black hole the coordinatelocation of its center, its conformal mass ~M, conformal

spin ~S, and its boost velocity. We center the black holes onthe x axis at x ¼ d=2, use the same mass ~M ¼ 1 for bothblack holes, and set the boost velocity to ð0;d0=2; 0Þ.The conformal spins are always equal and are aligned withthe orbital angular momentum of the holes.In contrast to the CFMS data, there are now two parame-

ters that influence the black holes’ spins: (i) the rotation

parameterr in Eq. (40) and (ii) the conformal spin ~S. Forconcreteness, we choose to construct data for four different

values of the conformal spin: ~S= ~M2 ¼ 0, 0.5, 0.93, and0.99. For each choice, we construct a family of initial-datasets for different values of r, which we label as SKS-0.0,SKS-0.5, SKS-0.93, and SKS-0.99, respectively.Other choices that went into the construction of the SKS

initial-data sets are as follows:(i) The excision boundaries are chosen to be the coor-

dinate locations of the horizons of the individualKerr-Schild metrics; i.e., they are surfaces of con-stant Kerr radius

rKerrexc ¼ ~rþ :¼ ~Mþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~M2 ~S2

p; (70)

length-contracted by the Lorentz factor appropriatefor the boost velocity of each black hole. This lengthcontraction accounts for the tangential motion of thehole but neglects the much smaller radial motion.

(ii) When superposing the individual Kerr-Schild met-rics, we use a damping length scale w ¼ 10rKerrexc [cf.Eqs. (45) and (46)], except for the SKS-0.99 family,which uses w ¼ d=3.

(iii) The orbital frequency 0 and radial expansion _a0are held fixed along each family. We expect that ourchoices for 0 and _a0 will lead to bounded, fairlycircular orbits, since


084017-14

jEADM MKjEADM

& 3%: (71)

In Sec. VB we reduce the orbital eccentricity forone data set in the family SKS-0.93.

We again solve the XCTS equations using the spectralelliptic solver of Ref. [62]; the families of SKS initial-datasets that we construct are summarized in Table I. Theelliptic solver needs some initial guess for the variablesto be solved for; we superpose the respective single-black-hole Kerr-Schild quantities, i.e.,

c ¼ 1; (72a)

c ¼ 1þ Xna¼1

er2a=w2aða 1Þ; (72b)

i ¼ Xna¼1

er2a=w2ai

a; (72c)

where n ¼ 2 and a and ia are the lapse and shift,

respectively, corresponding to the boosted, spinning Kerr-Schild metrics gaij used in the conformal metric ~gij.

Convergence of the elliptic solver and spin is demonstratedin Fig. 10 by showing the decreasing constraint violation8

and differences in spin with increasing resolution.We now turn our attention to the physical properties of

the SKS initial-data sets. Figure 11 shows the horizon massM and the dimensionless spin of either black hole for thefour families of SKS initial data. As expected, we find thatgenerally the spin increases with increasingr. For eachof the SKS families, we find that the elliptic solver fails toconverge for sufficiently larger. We suspect that the SKSfamilies exhibit a turning point, similar to the CFMS-singleand binary-black-hole initial data shown in Figs. 7 and 9. Ifthis is the case, Fig. 11 only shows the lower branch of eachfamily, and an additional branch of solutions will bepresent. Because we are satisfied with the spin magnitudesthat are possible along the lower branch, we do not attemptto find the upper branch here.

In contrast to the CFMS data sets (where the lowerbranch only allowed spins as large as & 0:85), the SKSinitial data allow spins that are quite close to unity. For thedifferent SKS families, we are able to construct initial datawith spins as large as

(i) 0:95 for SKS-0,(ii) 0:985 for SKS-0.5,(iii) 0:998 for SKS-0.93,(iv) 0:9997 for SKS-0.99.

These spins are far closer to extremal than possible withBowen-York initial data [ & 0:984 (Fig. 2)] or confor-mally flat, maximally sliced XCTS initial data [ & 0:85

40 60 80

N1/3

10-10

10-8

10-6

10-4

10-2

12 16 20L

AH

SKS-HeadonSKS-0.93-E3SKS-0.99, Ω

r=0.378

χ(LAH

)-χ(LAH

-2)Constraint violation

FIG. 10 (color online). Convergence of the spectral ellipticsolver. Left panel: The residual constraint violation as a functionof the total number of grid points N when running the ellipticsolver at several different resolutions. Right panel: Convergenceof the black-hole dimensionless spin [Eq. (1)] with increasingresolution LAH of the apparent horizon finder, applied to thehighest-resolution initial-data set of the left panel. The threecurves in each panel represent three different initial-data sets:one from the family SKS-0.99, as well as the two initial-data setsthat are evolved in Sec. V.

0.34 0.36 0.380.992

0.996

1

0.9

1.2

1.5

1.8

Mas

s M

SKS-0.0SKS-0.5SKS-0.93SKS-0.99SKS-0.93-E3SKS-Headon

0 0.1 0.2 0.3 0.4Ω

r

0

0.2

0.4

0.6

0.8

1

Spin

χ

FIG. 11 (color online). The mass M (upper panel) and dimen-sionless spin (lower panel) of one of the holes for superposedKerr-Schild, binary-black-hole initial-data sets with spinsaligned with the orbital angular momentum. The mass andspin are plotted against r [Eq. (40)] for four different choicesof the conformal spin: ~S ¼ 0, 0.5, 0.93, and 0.99. Also shown arethe data sets SKS-0.93-E3—identical to the r ¼ 0:28 ~M, ~S ¼0:93 ~M2 data set on the solid curve but with lower eccentricity—and SKS-Headon; both sets are evolved in Sec. V. The inset inthe lower panel shows a close-up of the spins as they approachunity, with symbols denoting the individual data sets.

8The constraint violation isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikCk2L2 þ kCkiL2kCkjL2ij

q, where C

and Ci are the residuals of Eqs. (10) and (11) and the L2 norm isgiven by Eq. (73).


084017-15

or & 0:99 along the lower and upper branch, respectively(Fig. 7)].

We note that the spins in the SKS binary-black-holeinitial-data families are only weakly dependent on theorbital parameters 0 and _a0. This can be seen from theindividual data point labeled SKS-0.93-E3 shown inFig. 11. This data set uses different values for 0 and _a0but is nevertheless close to the family SKS-0.93. Theinitial-data sets SKS-0.93-E3 and SKS-Headon will bediscussed in detail in Sec. V.

The inset of Fig. 11 highlights a remarkable feature ofthe SKS-0.93 and SKS-0.99 families: With increasing r,the spin initially increases but eventually decreases.Figure 12 investigates this behavior in more detail, wherethis effect is more clearly visible in the lower two panels:Both the spin and the extremality parameter of theapparent horizon change direction and begin to decrease.Forr smaller than this critical value, the apparent horizonfinder always converges onto the excision surfaces, whichby virtue of the boundary condition Eq. (39) are guaranteedto be marginally trapped surfaces. As r is increasedthrough the critical value (at which and change direc-tion), a secondmarginally trapped surface (solid line) splits

off from the excision surface (dashed line) and movescontinuously outward. This can be seen in the upper panelsof Fig. 12, which plot the minimal and maximal coordinateradius and the irreducible mass of both the excision surfaceand the outermost marginally trapped surface, which is bydefinition the apparent horizon.But what about the excision surface? The boundary

condition Eq. (39) forces the excision surface to be amarginally trapped surface, independent of the value ofr. For sufficiently larger, however, the excision surfaceis surrounded by a larger marginally trapped surface andthus is not the apparent horizon. The dashed lines in Fig. 12present data for the excision surface. These lines continuesmoothly across the point where the second marginallytrapped surface forms. The extremality parameter forthe excision surface continues to increase and eventuallybecomes larger than unity; the excision surface can then bethought of as having a superextremal spin. However, forthe outer marginally trapped surface—the true apparenthorizon—the extremality parameter always satisfies <1. The irreducible massMirr of this surface increases fasterthan the spin, and therefore ¼ S=ð2M2

irrÞ decreases withincreasing r.One might interpret these results as support of the cos-

mic censorship conjecture. The XCTS boundary condi-tions (39) and (40) control the location and the spin ofthe excision surface. By appropriate choices for the shiftboundary condition (40), we can force the excision surfaceto become superextremal. However, before this can hap-pen, a new horizon appears, surrounding the excisionsurface and hiding it from ‘‘our’’ asymptotically flat endof the spacetime. The newly formed outer horizon alwaysremains subextremal.

C. Suitability for evolutions

In the previous sections, we have constructed a widevariety of binary-black-hole initial-data sets. To get someindication about how suitable these are for evolutions, weconsider the initial time derivatives of these data sets @tgijand @tKij. Recall that solutions of the XCTS equations give

a preferred initial lapse and shift for the evolution of theinitial data; hence, the time derivatives @tgij and @tKij can

be computed by simply substituting the initial data into theADM evolution equations. We expect initial data withsmaller time derivatives to be closer to quasiequilibriumand to have less initial spurious radiation.Figure 13 presents the L2 norms of the time derivatives

k@tgijkL2 and k@tKijkL2 where the L2 norm of a tensor

TijkðxÞ evaluated at N grid points xi is defined as

kTijkkL2 :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XNi¼0

T2ðxiÞvuut ; (73)

where

0.37 0.38 0.39Ω

r

0.32 0.33 0.34 0.35Ω

r

0.85

0.90

0.95

1.00

1.05

1.0

1.2

1.4

1.6

1.8

χ

ζ

Mirr

min radius

max radius

Apparent horizon ID excision boundary

ζ

χ

Mirr

min radius

max radius

SKS-0.93 SKS-0.99

FIG. 12 (color online). The irreducible mass Mirr andEuclidean coordinate radius r (upper panels) and dimensionlessspin :¼ S=M2 and spin-extremality parameter :¼ S=ð2M2

irrÞ(lower panels) for one of the black holes in the SKS-0.93 (left)and SKS-0.99 (right) initial-data-set families. These quantitiesare computed on two surfaces: (i) the apparent horizon (solidlines) and (ii) the excision boundary of the initial data (dashedlines). Because we enforce that the excision surface is a mar-ginally trapped surface, typically the apparent horizon and ex-cision boundary coincide. However, if r is increased beyondthe values where approaches unity, the apparent horizon liesoutside of the excision surface. The excision surface can obtainsuperextremal spins ( > 1) but only when it is enclosed by asubextremal horizon.


084017-16

T :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTijk...Ti0j0k0...

ii0jj0kk0 . . .q

: (74)

Figure 13 shows that generally @tKij is larger than @tgij.

This has also been found in previous work, e.g., [63], and isnot surprising, because the XCTS formalism allows somecontrol over the time derivative of the metric through thefree data ~uij ¼ @t~gij, whereas there is less control of @tKij.

We note that for CFMS data, the time derivatives are largerand grow more rapidly with than for SKS data; inparticular, the time derivatives on the upper branch are10 times larger than for SKS-initial data, suggestingthat these data are much farther from equilibrium.

In the SKS case, the time derivatives of Kij have local

minima at particular values ofr; comparison with Fig. 11gives spins at these minima of k@tKijkL2 as follows:

(i) SKS-0.5: r 0:1, 0:45;(ii) SKS-0.93: r 0:28, 0:93;(iii) SKS-0.99: r 0:34, 0:98.

Note that these minima occur at values ofr such that ~S= ~M2; that is, transients in the initial data and presumablythe spurious radiation are minimized when the conformalspin and AKV spin are consistent. For this reason, we

conclude that SKS initial data with ~S= ~M2 is prefer-able; this is the type of initial data we will evolve in thenext section.

Also note that minimizing the spurious radiation haspurely numerical advantages: The spurious radiation typi-

cally has finer structure (and thus requires higher resolu-tion) than the physical radiation. If such radiation isminimized, the numerical evolutions may require lessresolution and will be more efficient. Conformally curvedinitial data has been found to reduce the amount of spu-rious radiation in Refs. [39,64].

V. EXPLORATORY EVOLUTIONS OF SKS INITIALDATA

So far, we have confined our discussion to black-holespins in the initial data. In this section, we compare theinitial spin to the value to which the spin relaxes after theinitial burst of spurious radiation, when the holes havesettled down. Recall, for instance, that for Bowen-Yorkpuncture initial data with spins close to the maximal pos-sible value [ðt ¼ 0Þ 0:98], the spins quickly relax byabout 0:05 to a maximal possible relaxed value ofðtrelaxÞ 0:93 (cf. [25]). While the SKS data presented inSec. IVB can achieve larger initial spins [ðt ¼ 0Þ ¼0:9997] than conformally flat puncture data, only evolu-tions can determine and ðtrelaxÞ.Therefore, in this section we perform brief, exploratory

evolutions of some SKS initial-data sets to determine for those data sets.9 Besides determination of ðtrelaxÞ,these evolutions will also allow us to demonstrate thatthe technique of eccentricity reduction developed inRef. [49] is applicable to SKS initial data as well as tocompare the spin measures defined in Appendixes A and B.The focus here lies on initial data, and we evolve only longenough for our purposes. Longer simulations that continuethrough merger and ringdown are the subject of ongoingresearch.This section is organized as follows. In Sec. VA, we

summarize the evolution code that we will use. In Sec. VB,we perform eccentricity reduction on one of the data sets inthe SKS-0.93 family, which corresponds to an orbitingbinary black hole with equal masses and equal spins (ofmagnitude 0:93) aligned with the orbital angular mo-mentum. Then, in Sec. VC, we evolve the resulting low-eccentricity data set (labeled SKS-0.93-E3). Finally, inSec. VD, we evolve a head-on plunge of SKS initial data(labeled SKS-Headon) representing two widely separatedblack holes with initial spins of magnitude ¼ 0:970 anddirection normal to the equatorial plane.

A. Description of evolution code

The initial data are evolved using the Caltech-Cornellpseudospectral evolution code SpEC [65]. The details ofthe evolution methods, equations, and boundary conditionsthat we use are the same as those described in Ref. [66].The singularities are excised, with the excision surfaces

0 0.1 0.2 0.3 0.4Ω

r

10-4

10-3

10-2

10-1

100

101

EA

DM

||d/

dt g

ij|| L2

0 0.1 0.2 0.3 0.4Ω

r

10-4

10-3

10-2

10-1

100

101

EA

DM

2 ||

d/dt

Kij|| L

2

CFMSSKS-0.0SKS-0.5SKS-0.93SKS-0.99SKS-0.93-E3SKS-Headon

FIG. 13 (color online). The time derivatives of the metric (leftpanel) and extrinsic curvature (right panel). In the SKS data sets,k@tKijkL2 has minima near values of r for which the dimen-

sionless spin is approximately equal to the spin ~S of theconformal metric (cf. Fig. 11). On the upper branch of theCFMS excision data, where the spin is > 0:83 (Fig. 9), thetime derivatives become much larger than the SKS time deriva-tives. The data sets SKS-0.93-E3 (with ~S ¼ 0:93) and SKS-Headon (with ~S ¼ 0:97) are evolved in Secs. VC and VD;the time derivatives are significantly lower for the set SKS-Headon because of the larger coordinate separation of the holes(d ¼ 100 vs d ¼ 32).

9Note that there is no universal value of —it will differ fordifferent initial-data sets, even within the same family of initialdata.


084017-17

chosen to lie slightly inside the black-hole horizons. Notethat whereas Ref. [66] excises coordinate spheres insidethe black holes’ apparent horizons, here we use Lorentz-contracted ellipsoidal excision boundaries which areadapted to the shape of the initial apparent horizons.

The highest-resolution initial-data set (with N 853

grid points) is interpolated onto evolution grids labeledN1, N2, and N3 with approximately 613, 673, and 743

grid points, respectively. The outer boundary is at a coor-dinate radius of r ¼ 32d for the orbiting simulation dis-cussed in Secs. VB and VC and at r ¼ 14d for the head-onsimulations discussed in Sec. VD. This translates to aboutr ¼ 450EADM and r ¼ 620EADM for the orbiting and head-on simulations, respectively. As in earlier simulations[49,65,66], a small region of the evolution grid lies insidethe horizon and is not covered by the initial-data grid; weextrapolate c , c , and i into this region and then com-pute gij and Kij.

B. Eccentricity removal for orbiting SKS binaries

We obtain initial data with small orbital eccentricityusing the iterative method of Ref. [49], as refined inRef. [66], applied here for the first time to binary-black-hole data with rapid spin. In this method, the choice of 0

and _a0 for the next iteration are made so that if the orbitwere Newtonian, the eccentricity would vanish. For thenon-Newtonian orbit here, successive iterations succeed inreducing the orbital eccentricity.

This procedure is based on the proper separation sbetween the apparent horizons, measured along a coordi-nate line connecting the geometric centers of the apparenthorizons. The time derivative ds=dt is fitted to a five-parameter curve that, together with the initial proper sepa-ration sðt ¼ 0Þ, is used to define the eccentricity e and todefine improved values for 0 and _a0. Specifically,

ds

dt:¼ A0 þ A1tþ B cosð!tþ ’Þ; (75a)

e :¼ B

!sðt ¼ 0Þ ; (75b)

0;new :¼ 0 þ B sin

2sðt ¼ 0Þ ; (75c)

_a0;new :¼ _a0 B cos

sðt ¼ 0Þ : (75d)

Heuristically, the eccentricity is embodied by the oscillat-ing part of ds=dt.

Figure 14 illustrates the eccentricity reduction for one ofthe data sets in family SKS-0.93. Plotted are the properseparation s and its derivative ds=dt for evolutions ofseveral initial-data sets (summarized in Table I):

(i) set SKS-0.93-E0, which is identical to the set infamily SKS-0.93 with r ¼ 0:28 (Fig. 11);

(ii) set SKS-0.93-E1, which is the same as SKS-0.93-E0except that the orbital frequency 0 is manually

adjusted to lower the orbital eccentricity somewhat;and

(iii) sets SKS-0.93-E2 and SKS-0.93-E3, which aresuccessive iterations (starting from set SKS-0.93-E1) of the eccentricity-reduction scheme Eqs. (75).

The ad hoc adjustment of 0 was somewhat effective,reducing e by about 50%. The subsequent iterations usingEqs. (75) reduced e by factors of about 5 and 8, respec-tively. Surprisingly, the lowest eccentricity, correspondingto a smooth inspiral trajectory, is obtained with a positive_a0 ¼ 3:332 104. This is not due to insufficient resolu-tion; for SKS-0.93-E3, we have verified that we obtain thesame eccentricity e 0:001 for all three numerical reso-lutions N1, N2, and N3.Note that we choose to stop the evolutions at about t ¼

670EADM, which corresponds to about 1.9 orbits; this issufficient for reducing the eccentricity and for measuring. In the next subsection, we discuss the evolution of thelow-eccentricity set SKS-0.93-E3 in detail, focusing on therelaxation of the spin .

C. Low-eccentricity inspiral with 0:93

We evolved the data set SKS-0.93-E3 at three differentnumerical resolutions for a duration of about 670EADM,corresponding to about 1.9 orbits. From post-Newtoniantheory [67], we estimate that this simulation would proceedthrough about 20 orbits to merger.Figure 15 presents a convergence test for this run. The

lower panel of Fig. 15 shows the normalized constraintviolation [see Eq. (71) of Ref. [68] for the precise defini-tion]. While the constraints are small, the convergenceseems poor until t 500EADM. For this time period the

0 200 400 600t / E

ADM

-0.02

-0.01

0

0.01

0.02

ds/d

t

13

14

15

16

17

s / E

AD

M

SKS-0.93-E0 (e~0.06)SKS-0.93-E1 (e~0.04)SKS-0.93-E2 (e~0.008)SKS-0.93-E3 (e~0.001)

FIG. 14 (color online). Eccentricity reduction for evolutions ofsuperposed Kerr-Schild binary-black-hole initial data. Theproper separation s (upper panel) and its time derivative ds=dt(lower panel) are plotted for initial-data sets SKS-0.93-E0,-E1, -E2, and -E3, which have successively smaller eccentricitiese. All evolutions are performed at resolution N1.


084017-18

constraint violations at high resolution N3 are dominatedby the outgoing pulse of spurious radiation—i.e., far awayfrom the black holes—which we have not attempted toadequately resolve. At t 500EADM, the pulse of spuriousradiation leaves the computational domain through theouter boundary; afterwards, the constraints decrease expo-nentially with increasing resolution, as expected.

The upper panel of Fig. 15 shows the AKV spin AKV ¼S=M2 for the three runs with different resolutions N1, N2,and N3. Based on the difference between N2 and N3, thespin of the evolution N3 should be accurate to a few partsin 104. For the time interval 5< t=EADM < 670, the mea-sured spin on resolution N3 is consistent with begin con-stant within its estimated accuracy. Very early in thesimulation, t < 5EADM, the spin changes convergentlyresolved from its initial value ðt ¼ 0Þ ¼ 0:927 48 to arelaxed value ðtrelaxÞ ¼ 0:927 14 (see inset of Fig. 15).Therefore, for SKS-0.93-E3, we find ¼ 0:000 34.

Contrast this result with the evolution of a binary-black-hole puncture initial-data set with large spins, which isreported in Ref. [25]: For that particular evolution, ðt ¼0Þ ¼ 0:967, ðtrelaxÞ ¼ 0:924, i.e., ¼ 0:043, more thana factor 100 larger than for the evolution of SKS-0.93-E3reported here. This comparison is somewhat biased againstthe puncture evolution in [25], which starts at a smallerseparation possibly resulting in larger initial transients.However, even in the limit that the black holes are infi-nitely separated (i.e., in the single-black-hole limit), thespins in Bowen-York puncture data relax to values near"J ¼ JADM=E

2ADM; to achieve a final spin of ðtrelaxÞ

0:93, the initial spin of Bowen-York data must be ðt ¼0Þ 0:98 (cf. Fig. 2 of Ref. [25]). We conclude that the

spin relaxes by a much smaller amount in the SKS casethan in Bowen-York puncture or inversion-symmetric data.Figure 15 and the discussion in the previous paragraph

only address the behavior of the AKV spin, where theapproximate Killing vectors are computed from the mini-mization problem [cf. Eq. (A10)]. We now compare thedifferent spin definitions we present in Appendixes A andB. Figure 16 compares these different definitions of theblack-hole spin for the N3 evolution of initial-data setSKS-0.93-E3. Shown are the AKV spin of one hole inthe binary, the scalar-curvature (SC) spins min

SC and maxSC

of Appendix B [Eqs. (B2a) and (B2b)], and also the spinobtained by using Eq. (A1) with a coordinate rotationvector instead of an approximate Killing vector (whichwe call the ‘‘coordinate spin’’ here). After the holes haverelaxed, the SC spins track the AKV spin more closely thandoes the coordinate spin. However, during very early times,as the holes are relaxing and the horizon shape is verydistorted, the SC spins show much larger variations.Consequently, the SC spin is a poorer measure of thespin at early times than even the coordinate spin.

D. Head-on plunge with 0:97

In the previous subsection, we have seen that, for SKSbinary-black-hole initial data with ¼ 0:93, the initialspins change by only a few parts in 104. A spin 0:93is roughly the largest possible equilibrium spin that isobtainable using standard conformally flat, Bowen-Yorkpuncture data (cf. the discussion at the beginning ofSec. V). We now begin to explore binary-black-hole simu-lations with spin magnitudes that are not obtainable withBowen-York initial-data methods.

0.927

0.928

0.929χ A

KV

0 200 400 600t / E

ADM

10-6

10-5

10-4

Con

stra

ints

N1N2N3

0 5

0.9272

0.9274

FIG. 15 (color online). Convergence test of the evolution ofthe initial-data set SKS-0.93-E3. Shown are evolutions on threedifferent resolutions N1, N2, and N3, with N3 being the highestresolution. The top panel shows the AKV spin of one of the holesas a function of time, with the top inset showing the spin’s initialrelaxation; the bottom panel shows the constraint violation as afunction of time.

0 10 20 3010-8

10-6

10-4

10-2

0.928

0.930

0.932

Spin

100 200 300 400 500 600

χAKV

χcoord

χSCmin

χSCmax

t / EADM

|χA

KV

- χ

X| /

χA

KV

FIG. 16 (color online). A comparison of different definitionsof the spin. The top panel shows the spin as a function of time forseveral different measures of the spin; the bottom panel showsthe fractional difference between AKV and alternative spindefinitions. Note that for t < 30EADM, the time axis has a differ-ent scaling to make the initial transients visible.


084017-19

We construct and evolve SKS binary-black-hole data fora head-on plunge of two equal-mass black holes with spinsof equal magnitude ¼ 0:97 and with the spins orthogo-nal to the line connecting the black holes. This data set,labeled SKS-Headon, is summarized in Table I and wasbriefly discussed in Sec. IVB; cf. Figs. 10, 11, and 13. Asfor the orbiting evolution SKS-0.93-E3, we adjust the

rotation parameter r so that conformal spin ~S= ~M2 andAKV spin are approximately equal. Starting such asimulation at close separation results in rapid coordinatemotion of the apparent horizons during the first few EADM

of the evolution. These motions are currently difficult totrack with our excision code; therefore, we begin at a largerseparation d than we used in the nearly circular data setsdescribed previously.

Figure 17 presents a convergence test of the constraints(lower panel) and the AKV spin AKV (upper panel) duringthe subsequent evolution. Again, we are interested in theinitial relaxation of the spins; therefore, we choose to stopevolution at t 120EADM. During this time, the black-holeproper separation decreased from sðt ¼ 0Þ ¼ 47:6EADM tosðt ¼ 120Þ ¼ 44:1EADM.

During the first 10EADM, AKV shows (a numericallyresolved) decrease of about 3 105; this change arisesdue to initial transients as the black holes and the fullgeometry of the spacetime relax into an equilibrium con-figuration. Subsequently, the spin remains constant towithin about 104, where these variations are dominatedby numerical truncation error.

Figure 18 compares our various spin measures for thehead-on simulation. Interestingly, the spin coord computedfrom coordinate rotation vectors agrees much better withAKV than for the SKS-0.93-E3 evolution, perhaps becausethe black holes here are initially at rest. The SC spins min

SC

and maxSC , derived from the scalar curvature of the apparent

horizon [Eqs. (B2a) and (B2b)], show some oscillations atearly times; after the initial relaxation, the SC spin agreeswith the AKV spin to about 1 part in 104.

VI. DISCUSSION

A. Maximal possible spin

In this paper, we have examined a variety of methods forconstructing black-hole initial data with a particular em-phasis on the ability to construct black holes with nearlyextremal spins. These are spins for which the dimension-less spin ¼ S=M2 and spin-extremality parameter ¼S=ð2M2

irrÞ are close to unity.

When discussing black-hole spin, one needs to distin-guish between the initial black-hole spin and the relaxedspin of the holes after they have settled down. Usingconformally flat BY data (both puncture data orinversion-symmetric data) for single black holes, the larg-est obtainable spins are 0:984, 0:833 (cf.Ref. [61] and Fig. 2). With CFMS, QE-XCTS data, weare able to obtain initial spins as large as 0:99, 0:87 for single black holes (Fig. 7). The limitations of BYpuncture data and CFMS QE-XCTS data are alreadypresent when constructing highly spinning single blackholes; therefore, we expect the methods to be able toconstruct binary-black-hole data with similar spins as forsingle holes—i.e., up to about 0.98. Construction of CFMSQE-XCTS binary-black-hole initial data confirms this con-jecture (compare Fig. 9 with Fig. 7).For SKS initial data, the situation is different. For single

black holes, SKS data reduce to the analytical Kerr solu-

0 20 40 60 80 100 120t / E

ADM

10-5

10-4

10-3

Con

stra

ints

N1N2N3

0.97008

0.97012

0.97016

χ AK

V

FIG. 17 (color online). Convergence test of the head-on evo-lution SKS-Headon. Shown are evolutions at three differentresolutions N1, N2, and N3, with N3 being the highest resolu-tion. The top panel shows the AKV spin of one of the holes as afunction of time; the bottom panel shows the constraint viola-tions as a function of time.

0 20 40 60 80 100 120t / E

ADM

10-8

10-6

10-4

10-2

|χA

KV

- χ

X| /

χA

KV

0.968

0.970

0.972

Spin

χAKV

χcoord

χSCmin

χSCmax

FIG. 18 (color online). A comparison of various measures ofthe spin for the head-on evolution of data set SKS-Headon,which is a plunge of two equal-mass black holes with parallelspins of magnitude AKV ¼ S=M2 ¼ 0:970 pointed normal tothe equatorial plane. The top panel shows various measures ofthe spin as a function of time, and the bottom panel shows thefractional difference between the AKV spin AKV and alterna-tive spin definitions.


084017-20

tion, without any limitations on the spin magnitude. Thuslimitations of SKS data will only be visible for binary-black-hole configurations. As Secs. IV and V show, how-ever, those limitations are quite minor. SKS data can in-deed achieve initial spins that are much closer toextremality than what is possible with BY data or CFMSQE-XCTS data; we have explicitly demonstrated this byconstructing SKS data for binary black holes with 0:9997, 0:98, as can be seen from Figs. 11 and 12.

As the black-hole spacetimes settle into equilibrium andemit spurious gravitational radiation, the initial spin decreases to a smaller relaxed spin ðtrelaxÞ. Thus an inter-esting quality factor for high-spin black-hole initial data is ¼ ðt ¼ 0Þ ðtrelaxÞ [Eq. (3)] considered as a func-tion of the relaxed spin. The magnitude of is indicativeof the amplitude of any initial transients, whereas themaximally achievable ðtrelaxÞ gives the largest possiblespin which can be evolved with such initial data. Figure 19presents this plot, with the circle and cross representing thetwo evolutions of SKS data which were described inSec. V.

We have not evolved high-spin puncture data nor high-spin CFMS-XCTS data; therefore, we do not know pre-cisely for these initial data. We estimate forpuncture data by noting that evolutions of single-hole,BY puncture data with large spins show [25] that theblack-hole spin :¼ S=M2 relaxes approximately to theinitial value of "J :¼ JADM=E

2ADM. Therefore, for BY

puncture data, we approximate

"J ðt ¼ 0Þ; (76)

ðtrelaxÞ "J: (77)

This curve is plotted in Fig. 19. Because high-spin single-black-hole, CFMS QE-XCTS initial data and BY puncturedata have quite similar values of ðt ¼ 0Þ and "J, as wellas similar embedding diagrams (cf. Fig. 8), we conjecturethat Eqs. (76) and (77) are also applicable to CFMS QE-XCTS data. This estimate is also included in Fig. 19. Wesee that both types of initial data result in a of similarmagnitude which grows rapidly with relaxed.Perhaps the most remarkable result of Fig. 19 is the

extremely small change in black-hole spin during therelaxation of SKS initial data, even at spins as large as ¼0:97. The small values of combined with the ability toconstruct initial data with initial spins ðt ¼ 0Þ as large as0.9997 (cf. Fig. 11) makes it highly likely that SKS initialdata are capable of constructing binary black holes withrelaxed spins significantly closer to unity than 0.97.Evolutions of initial data with spins much closer to unity,i.e., farther into the regime that is inaccessible to confor-mally flat data, are a subject of our ongoing research.In summary, the two main results of this paper are as

follows:(i) SKS initial data can make binary black holes that

initially have nearly extremal spins, and(ii) for SKS initial data, the relaxed spin is quite close to

the initial spin, even when the spin is large.

B. Additional results

While working toward the main results discussed in theprevious subsection, we have also established several addi-tional interesting results. We have considered spinning,single-black-hole, puncture data which is identical tosingle-black-hole, spinning, inversion-symmetric data.Using this correspondence and our accurate spectral ellip-tic solver, we revisited the relation between black-hole spin, specific total angular momentum of the spacetime "J,and the spin parameter S for BY puncture data and estab-lished in Fig. 2 that both and "J approach their limits forS ! 1 as power laws; cf. Eqs (49) and (50). We have alsoextended the analytical analysis of Dain, Lousto, andZlochower [25] of the throat region of high-spin puncturedata toward more quantitative results, including the preciseamplitudes of the conformal factor, throat circumferenceand throat length, as well as their scaling with spin pa-rameter S and puncture mass mp [Eqs. (54)–(58)].

Furthermore, Ref. [25] implicitly assumed that the throatregion is approximately spherically symmetric; our Fig. 5presents explicit evidence in support of this assumption butalso shows that the throat is not precisely sphericallysymmetric.We have also examined high-spin QE-XCTS initial data

employing the common approximations of CFMS. With

0.5 0.6 0.7 0.8 0.9 1.0

0.0001

0.001

0.01

0.1

BY puncture dataCFMS QE-XCTS dataSKS-0.93-E3SKS-0.97-HeadOn

∆χ

χ(trelax

)

FIG. 19 (color online). The change in black-hole spin during the initial relaxation of black-hole initial data plotted as afunction of the black-hole spin after relaxation. The SKS initialdata constructed in this paper have smaller transients and allowfor larger relaxed spins.


084017-21

increasing angular frequency r of the horizon, we dis-cover nonunique solutions. Thus, the nonuniqueness of theXCTS equations can be triggered not only by volume terms(as in [34]) but also through boundary conditions [in thiscase, by Eq. (40)]. Interestingly, CFMS QE-XCTS dataappear to be very similar to BY puncture data, in regard tonearly extremal spins. Both data formalisms result in simi-lar maximal values of ðt ¼ 0Þ and "J (Figs. 2, 7, and 19)and have embedding diagrams which develop a length-ening throat as the spin is increased (Fig. 8).

We also have found an interesting property of the hori-zon geometries for SKS data, which one might interpret assupport of the cosmic censorship conjecture. Specifically,we find that by increasing r sufficiently we can in factforce the excision boundaries of the initial data to be‘‘horizons’’ (i.e., marginally trapped surfaces) with super-extremal spin ( > 1). However, these superextremal sur-faces are always enclosed by a larger, subextremal ( < 1)apparent horizon.

To measure black-hole spins, we have employed andcompared several different techniques to measure black-hole spin. Primarily, we use a quasilocal spin definitionbased on (approximate) Killing vectors [Eq. (A1)]. Thisformula requires the choice of an approximate Killingvector, and we have used both straightforward coordinaterotations to obtain coord and solved Killing’s equation in aleast-squares sense to obtain AKV (see Appendix A fordetails). Furthermore, we introduced a new technique todefine black-hole spin which does not require choice of anapproximate Killing vector and is invariant under spatialcoordinate transformations and transformations associatedwith the boost gauge ambiguity of the dynamical horizonformalism. This new technique is based on the extrema ofthe scalar curvature of the apparent horizon. Figures 16 and18 show that all four spin measures agree to good preci-sion, but differences are noticeable. The spin measuresbased on the horizon curvature exhibit more pronouncedvariations during the initial transients, and the quasilocalspin based on coordinate rotations is off by several tenthsof a percent. The quasilocal spin based on approximateKilling vectors AKV has the smallest initial variations.

Finally, we would like to point out that a modifiedversion of the SKS-initial data has been very successfullyused to construct black-hole–neutron-star initial data [69].

ACKNOWLEDGMENTS

It is a pleasure to acknowledge useful discussions withIvan Booth, Gregory Cook, Stephen Fairhurst, LawrenceKidder, Lee Lindblom, Mark Scheel, Saul Teukolsky, andKip Thorne. The numerical calculations in this paper wereperformed using the Spectral Einstein Code (SpEC), whichwas primarily developed by Lawrence Kidder, HaraldPfeiffer, and Mark Scheel. We would also like to acknowl-edge the anonymous referee for reminding us of an im-portant technical caveat. Some equations in this paper were

obtained using MATHEMATICA. This work was supported inpart by grants from the Sherman Fairchild Foundation toCaltech and Cornell and from the Brinson Foundation toCaltech; by NSF Grants No. PHY-0652952, No. DMS-0553677, and No. PHY-0652929 and NASA GrantNo. NNG05GG51G at Cornell; and by NSF GrantsNo. PHY-0601459, No. PHY-0652995, and No. DMS-0553302 and NASA Grant No. NNG05GG52G at Caltech.

APPENDIX A: QUASILOCAL SPIN USINGAPPROXIMATE KILLING VECTORS (AKV SPIN)

In this appendix and the one that follows, we address thetask of defining the spin of a dynamical black hole, givengij and Kij. We use two different measures. The first,

defined here, is a standard quasilocal angular momentumdefined with approximate Killing vectors which corre-spond to approximate symmetries of a black hole’s hori-zon. The second measure, defined in Appendix B, infersthe spin from geometrical properties (specifically, from the

intrinsic scalar curvature R) of the apparent horizon, as-

suming that the horizon is that of a single black hole inequilibrium, (i.e., that the horizon is that of a Kerr blackhole). Note that quantities relating to the geometry of thetwo-dimensional apparent horizon surface H are denotedwith a ring above them, to avoid confusion with the analo-gous quantities on the spatial slice .It has become standard in the numerical relativity com-

munity to compute the spin angular momentum of a blackhole with the formula [70–72]

S ¼ 1

8

IH

isjKijdA; (A1)

where si is the outgoing normal of H embedded in and~ is an ‘‘azimuthal’’ vector field, tangent to H . The

azimuthal vector field ~ carries information about the‘‘axis’’ about which the spin is being computed. Thereare, however, far more vector fields on a two-sphere thanthere are axes in conventional Euclidean space. We mustfind suitable criteria for fixing these azimuthal vector fieldsin numerical simulations, so that they reduce to the stan-dard rotation generators when considered on a metricsphere.Because angular momentum is generally thought of as a

conserved charge associated with rotation symmetry—andindeed the quantity given in (A1) can be shown to be

conserved under time evolution [70,72] when ~ is aKilling vector of the dynamical horizon world tube—itmakes sense to consider Killing’s equation to be the essen-tial feature of the azimuthal vector field. If a Killing vectoron a dynamical horizon is a tangent to each (two-dimensional) apparent horizon, then the vector field mustbe a Killing vector of each apparent horizon. However in ageneral spacetime, on an arbitrary apparent horizon, thereis no reason to expect any Killing vectors to exist. So in the


084017-22

cases of most interest to numerical relativity, when thereare no true rotation symmetries, we must relax the sym-metry condition and find those vector fields that come‘‘closest’’ to generating a symmetry of the apparent hori-zon. In other words, we seek optimal approximate Killingvectors of the apparent horizon.

In [73], a practical method for computing approximateKilling vectors was introduced, which has since beenapplied on numerous occasions, e.g., [18,19,29,74]. Thismethod involves integrating the Killing transport equationsalong a predetermined network of coordinate paths. Theresulting vector field is guaranteed to be a Killing vectorfield if such a field exists and coincides with the computedfield at any point on the network. However if no trueKilling field exists, the integral of the Killing transportequations becomes path-dependent. This means that thecomputed vector field will depend in an essential way onthe network of paths chosen for the integral. Perhaps evenmore serious, if there is no true Killing field, then thetransport of a vector around a closed path will not neces-sarily be an identity map. As a result, the computed vectorfield cannot be expected to reduce to any smooth vectorfield in the limit that the network becomes more refined.This kind of approximate-Killing-vector field is simply notmathematically well-defined in the continuum limit.

Here we will describe a kind of approximate-Killing-vector field that, as well as having a well-defined contin-uum limit, is actually easier to construct than those of theKilling transport method, at least in our particular code.Our method is extremely similar to that described by Cookand Whiting [41] but was actually developed indepen-dently by one of the current authors [75].

1. Zero expansion, minimal shear

Killing’s equation

DðABÞ ¼ 0 (A2)

has two independent parts: the condition that ~ beexpansion-free

:¼ gABDAB ¼ 0 (A3)

and the condition that it be shear-free

AB :¼ DðABÞ 12gAB ¼ 0; (A4)

where uppercase Latin letters index the tangent bundle to

the two-dimensional surface, gAB is the metric on that

surface, and DA is the torsion-free covariant derivativecompatible with that metric.

When constructing approximate Killing vectors, a ques-tion arises: Which condition is more important, zero ex-pansion or zero shear? Shear-free vector fields (conformalKilling vectors) are simply coordinate rotation generatorsin the common case of coordinate spheres in a conformallyflat space. They are therefore readily available in that

context. Avery interesting and systematic approach to theiruse has been given by Korzynski [76], and they have beenused in the construction of conformally flat binary-black-hole initial-data sets [28,29]. However, in the case of ageneral surface in a general spatial slice, the conformalKilling vectors are not known a priori, and they are moredifficult to construct than expansion-free vector fields.Expansion-free vector fields have the additional benefitof providing a gauge-invariant spin measure on a dynami-cal horizon [70],10 so we restrict attention to the expansion-free case.Any smooth, expansion-free vector field tangent to a

topological two-sphere can be written as

A ¼ ABDBz; (A5)

where AB is the Levi-Civita tensor and z is some smoothpotential function.We assume that the function z has one local maximum,

one local minimum, and no other critical points. This is

equivalent to the assumption that the orbits of ~ are simpleclosed loops. In order for AA to have the proper dimen-sions, z must have dimensions of area. For the case of thestandard rotation generators of the metric two-sphere, thethree z functions are the three ‘ ¼ 1 spherical harmonics,multiplied by the square of the areal radius of the sphere.Within this space of expansion-free vector fields, we

would now like to minimize the following positive-definitenorm of the shear:

kk2 :¼IH

BCBCdA: (A6)

Substituting Eq. (A5) for ~ in this expression and integrat-ing twice by parts, kk2 takes the form of an expectationvalue:

kk2 ¼IH

zHzdA; (A7)

where H is the self-adjoint fourth-order differential opera-tor defined by

Hz ¼ D4zþ RD2zþDAR

DAz; (A8)

and D2 is the Laplacian on the (not necessarily round)

sphere, D4 is its square, and Ris the Ricci scalar curvature

of the sphere. In our sign convention, R ¼ 2 on the unit

sphere, so we can immediately see that Hz ¼ 0 when z is

10The dynamical horizon is essentially the world tube foliatedby the apparent horizons. The gauge freedom is that of extendingthe foliation off of this world tube. The fact that this gaugeinvariance occurs when ~ is expansion-free can most easily beshown by expressing the factor sjKij in Eq. (A1) in terms of theingoing and outgoing null normals to the two-surface. The boostfreedom in these null normals has no effect on the spin when ~ isexpansion-free.


084017-23

an ‘ ¼ 1 spherical harmonic and therefore that their asso-ciated vector fields are shear-free.

It is now tempting to minimize the functional kk2 in(A7) with respect to z. However, doing so will simplyreturn the condition that z lie in the kernel of H. If thereare no true Killing vectors, this will mean that z is a

constant and therefore that ~ vanishes. We need to restrictthe minimization procedure to cases that satisfy somenormalization condition. In this case, we require that thenorm of the vector field

IH

AAdA (A9)

take some given positive value. This restriction can bemade with the use of a Lagrange multiplier. Specifically,the functional we wish to minimize is

I½z :¼IH

zHzdAþ

IH

DAzDAzdA N

(A10)

for some yet undetermined positive parameter N. Note that is the Lagrange multiplier and we have made use of the

fact that Eq. (A5) implies that ~ ~ ¼ ~Dz ~Dz.Minimizing the functional I with respect to z returns ageneralized eigenvalue problem:

Hz ¼ D2z: (A11)

It is at this point that we can most easily clarify thedifference between our construction of approximateKilling vectors and that of Cook and Whiting in [41].The difference lies in the choice of norm in which theminimization problem is restricted. Rather than fixing thenorm (A9) to take some fixed value in the minimization,Cook and Whiting instead fix the dimensionless norm:I

HRAAdA: (A12)

In general, we see no particular reason to prefer eithernorm over the other, but for the current purposes we haveat least an aesthetic preference for (A9), which is positive-definite even at high spin, whereas (A12) is not, because

the scalar curvature Rof the horizon becomes negative

near the poles at high spin. If the norm (A9) in Eq. (A10) isreplaced by (A12), the result is the problem described in[41]:

Hz ¼ ðRD2zþDARDAzÞ: (A13)

In our numerical code, we discretize (A11) [or, option-ally, (A13), but not for any results published here] andsolve the resulting linear algebra problem with a LAPACK

routine [77]. Note, however, one technical peculiarity: Theoperators H and D2 in (A11) share a kernel, the space ofconstant functions. This means that this generalized eigen-

value problem is singular, a fact that can cause consider-able difficulties for the numerical solution [78]. The samecan be said of (A13). For our purposes, this complication iseasily evaded. Since we are working with a spectral code, itis easiest to discretize the problem using expansion into thespectral basis functions (coordinate spherical harmonics).When this is done, the space of constant functions—theshared kernel of the two operators—is simply the span of asingle basis function: the constant Y00. This basis functioncan easily be left out of the spectral expansion and therebyremoved from the numerical problem.Expansion into coordinate spherical harmonics has an-

other practical advantage. As noted earlier, for metricspheres in standard coordinates, the Killing vectors arisewhen z is given by an ‘ ¼ 1 spherical harmonic. Thus,assuming our horizon is nearly round, and noticeably so inthe given coordinates, the lowest basis functions (the ‘ ¼ 1spherical harmonics) should nicely approximate the in-tended eigenfunctions. The higher basis functions shouldsimply provide small corrections.In summary, the approach that we take to finding ap-

proximate Killing vectors begins with a spectral decom-position of Eq. (A11). This problem, of course, provides asmany eigenvectors as there are elements of the spectraldecomposition. We restrict attention to the three eigenvec-tors with smallest eigenvalues (ignoring the vector corre-sponding to the constant eigenfunction, which is physicallyirrelevant and removed from discretization), as these arethe ones corresponding to vector fields with the smallestshear, and, at least for spheres that are only slightly de-formed, the orbits of these vector fields are smooth closedloops.It must be noted that only the eigenvector with the

smallest eigenvalue corresponds to a vector field withstrictly minimum shear: Even locally, all other eigenvec-tors are saddle points of the minimization problem. Thethree of them taken together, however, provide a geomet-rically defined subspace of the vector space of expansion-free vector fields, a natural generalization of the rotationgenerators on metric spheres. Using these three vectorfields (normalized as described in the next subsection),one can define ‘‘components’’ of the spin angular momen-tum of a black hole11 and from these components infer thespin around an arbitrary axis or even a spin ‘‘magnitude’’using a metric on this three-dimensional space of general-ized rotation generators. In practice, we have found noneed to go quite so far. As mentioned in [41], the approxi-mate Killing vectors generally adapt themselves so well tothe horizon that one of the components is much larger thanthe other two, so this is considered the spin magnitude, and

11In fact, using the higher eigenvectors, one could in principlecompute higher-order multipole moments. We see this as anatural extension of the method laid out in [74] for definingthe higher multipole moments of axisymmetric black holes.


084017-24

the associated approximate Killing vector is considered todefine the spin axis.

2. Normalization

Solutions to the eigenproblem (A11) can only determinethe approximate Killing vectors up to a constant scaling.Fixing this scaling is equivalent to fixing the value of N in(A10). The standard rotation generators of metric spheresare normalized such that, when considered as differentialoperators along their various orbits, they differentiate withrespect to a parameter that changes by a value of 2 aroundeach orbit. Naively one would like to fix the normalizationof approximate Killing vectors in the same way, but asubtlety arises: We can only rescale the vector field by afixed, constant value. Rescaling differently along differentorbits would introduce extraneous shear and would removethe vector field from the pure eigenspace of (A11) in whichit initially resided. If an approximate-Killing-vector fieldhas different parameter circumferences around differentorbits, then it is impossible to rescale it such that theparameter distance is 2 around every orbit. The bestone can ask is that 2 is the average of the distancesaround the various orbits.

To consider this in detail, introduce a coordinate system,topologically the same as the standard spherical coordi-nates on the metric sphere but adapted to the potentialfunction z so that the latitude lines are the level surfacesof z (and, in particular, the poles are at the two criticalpoints we have assumed z to have). More precisely, choosez for the zenith coordinate on the sphere and an arbitraryrotational coordinate—say, the azimuthal angle in the en-compassing spatial slice, describing rotations about theaxis connecting the critical points of z—for the azimuthalcoordinate ’ on the sphere. If the parameter is defined

such that ~ ¼ ðd=d Þz¼const, then in the basis related to

these coordinates, the components of ~ are

zðz; ’Þ ¼dz

d

z¼const

¼ 0; (A14)

’ðz; ’Þ ¼d’

d

z¼const

: (A15)

Around a closed orbit CðzÞ, at fixed z, the parameter changes by a value of

ðzÞ ¼ZCðzÞ

d’

’ðz; ’Þ (A16)

¼ZCðzÞ

d’

’z@zz(A17)

¼ZCðzÞ

ffiffiffig

qd’; (A18)

where gis the determinant of the surface metric, evaluated

in the ðz; ’Þ coordinates. Note that Eq. (A18) follows fromEq. (A17) by the fact that the condition g

ABg

CD

ACBD ¼2 implies ’z ¼ 1=

ffiffiffig

q. The average value of , over the

various orbits, is

h i ¼ 1

zmax zmin

Z zmax

zmin

ZCðzÞ

ffiffiffig

qd’dz (A19)

¼ A

zmax zmin

; (A20)

where A is the surface area of the apparent horizon.Requiring this average to equal 2, we arrive at the nor-malization condition:

2ðzmax zminÞ ¼ A: (A21)

This normalization condition requires finding the mini-mum and maximum values of the function z, which is onlycomputed on a discrete grid. In our spectral code, inparticular, this numerical grid is quite coarse, so numericalinterpolation is needed, in combination with an optimiza-tion routine. We have implemented such routines to searchfor zmin and zmax, but a numerically cheaper normalizationcondition would be of interest. Such a condition ariseswhen one assumes that the black hole under considerationis approximately Kerr. In the Kerr metric, for the function zgenerating the true rotation generator of the Kerr horizon,the following identity holds:

IHðz hhziiÞ2dA ¼ A3

482; (A22)

where hhzii is the average of z over the sphere. The existenceof an identity of this form is somewhat nontrivial: The factthat the right side is given purely by the horizon area, andthat it does not involve the spin of the Kerr hole, is whatmakes this identity useful as a normalization condition.This normalization is much easier to impose and requiressignificantly less numerical effort.To close the discussion of spin computed from approxi-

mate Killing vectors, we demonstrate the effectiveness ofthe method in a simple test case: an analytic Kerr blackhole in slightly deformed coordinates. We begin with aKerr black hole of dimensionless spin parameter ¼ 1=2,in Kerr-Schild coordinates, but we rescale the x axis by afactor of 1.1. This rescaling of the x coordinate causes thecoordinate rotation vector x@y y@x to no longer be the

true, geometrical rotation generator. Indeed, when wecompute the quasilocal angular momentum (A1) on thehorizon using this coordinate vector, the result converges toa physically inaccurate value, as demonstrated by the blackdotted curve in Fig. 20. If, however, the approximateKilling vectors described above are used, the result is notonly convergent but physically accurate. Because the ac-curacy is slightly better with the normalization condition of


084017-25

Eq. (A22), that is the condition we use for all resultspresented in this paper.

APPENDIX B: SCALAR-CURVATURE SPIN

In this appendix, we define a spin measure in terms ofthe intrinsic geometry of the horizon, which we comparewith the AKV spin in Sec. V. The AKV spin described inAppendix A is a well-defined measure of black-hole spin,even when the holes’ horizons have only approximatesymmetries. At times sufficiently before or after the holesmerge, however, the horizons will not be too tidally dis-torted and thus will not be too different from the exactlyaxisymmetric horizons of Kerr black holes.

By assuming that the geometric properties of the horizonbehave precisely as they do for a Kerr black hole, one caninfer the hole’s spin from those properties. For instance, itis common to measure polar and equatorial circumferencesof the apparent horizon; the spin is then obtained by findingthe Kerr spacetime with the same circumferences [79–81].

To avoid introducing coordinate dependence by defining‘‘polar’’ and ‘‘equatorial’’ planes, we infer the spin from

the horizon’s intrinsic scalar curvature R. The horizon

scalar curvature Rhas previously been studied analytically

for Kerr-Newman black holes [82] and for Kerr black holes

perturbed by a distant moon [83]. Numerical studies of R

have focused attention on the quasinormal ringing ofsingle, perturbed, black holes [79] as well as on the shapeof the individual and common event horizons in Misner

data [84]. To our knowledge, the scalar curvature Rhas not

been previously used to infer the horizon spin in numericalsimulations.At a given point on a Kerr black hole’s horizon, the

horizon scalar curvature Rdepends only on the hole’s mass

M and spin S. The extrema of Rcan be expressed in terms

of the irreducible mass and dimensionless spin of the Kerrblack hole via Eqs. (1) and (2) as

minðRÞ ¼ 1þ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2

p2M2

irr

; (B1a)

maxðRÞ ¼ 2

M2irr

4ð2þ 2 þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2

qÞ: (B1b)

Solving for and requiring it to be real yields as a

function of Mirr and either minðRÞ or maxðRÞ. We takethese functions as definitions of the spin, even when thespacetime is not precisely Kerr:

ðminSC Þ2 :¼ 1 ½12 þM2

irr minðRÞ2; (B2a)

ðmaxSC Þ2 :¼ 2þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2M2

irr maxðRÞq

M2irr maxðRÞ

: (B2b)

The definitions of the spin given by Eqs. (B2a) and (B2b)are manifestly independent of spatial coordinates and arewell-defined for black holes that are tidally deformed.Also, as they only involve the intrinsic two-dimensionalgeometry of the apparent horizon, they are also manifestlyindependent of boost gauge, in the sense described in theprevious appendix.We expect min

SC and maxSC to be reasonable measures only

if tidal forces can be neglected. Tidal forces scale with thecube of the separation of the holes; for binary with holes ofequal massM and separation d, tidal coupling is negligible

when maxðRÞ minðRÞ M=d3.

We find it convenient to compute Rfrom (i) the scalar

curvature R associated with the three-dimensional metricgij of the spatial slice and (ii) the outward-pointing unit-

vector field si that is normal to H . This can by done bymeans of Gauss’s equation [e.g., Eq. (D.51) of Ref. [85](note that the Riemann tensor in Ref. [85] disagrees withours by an overall sign)]

R ¼ R 2Rijs

isj K2 þ K

ijK

ij; (B3)

where Rij and R were defined after Eq. (11) and where Kij

denotes the extrinsic curvature of the apparent horizon H

5 10 15L

AH

10-12

10-9

10-6

10-3

Err

or in

χ

χcoord

χSCmin

χSCmax

χAKV

(normalized by extrema)χ

AKV (normalized by integral)

FIG. 20 (color online). Error, relative to the analytic solution,of the spin on the horizon of a Kerr black hole in slightlydeformed coordinates. The vertical axis represents jcomputed analyticj, and data are shown for the spin computed with the

standard coordinate rotation vector (in deformed coordinates, sonot a true Killing vector) and with our AKVs using both theextremum norm Eq. (A21) and the integral norm Eq. (A22). Thespin computed from the coordinate rotation vector quickly con-verges to a physically inaccurate result. The spin from approxi-mate Killing vectors converges in resolution LAH to the correctvalue ¼ 1=2. Curves are also shown for the two spin measuresdefined in Appendix B. These spin measures also convergeexponentially to the physically correct result.


084017-26

embedded in (not to be confused with Kij, the extrinsic

curvature of the slice embedded in M). The horizonextrinsic curvature is given by

K

ij ¼ risj siskrksj: (B4)

Inserting Eq. (B4) into Eq. (B3) shows that R

can beevaluated exclusively in terms of quantities defined onthe three-dimensional spatial slice .

The accuracy of these spin measures is demonstrated inFig. 20, which shows a Kerr black hole with ¼ 1=2 inslightly deformed coordinates so that the coordinate rota-tion vector no longer generates a symmetry. Again, bothminSC and max

SC converge exponentially to the physically

accurate result.

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084017 (2008) binary-black-hole initial data with nearly

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